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How to Solve Simultaneous Equations Using Substitution Method
Last Updated: October 10, 2022 References
This article was co-authored by wikiHow Staff . Our trained team of editors and researchers validate articles for accuracy and comprehensiveness. wikiHow's Content Management Team carefully monitors the work from our editorial staff to ensure that each article is backed by trusted research and meets our high quality standards. This article has been viewed 188,749 times. Learn more...
Simultaneous equations are two linear equations with two unknown variables that have the same solution. Solving equations with one unknown variable is a simple matter of isolating the variable; however, this isn’t possible when the equations have two unknown variables. By using the substitution method, you must find the value of one variable in the first equation, and then substitute that variable into the second equation. [1] X Research source While it involves several steps, the substitution method for solving simultaneous equations requires only basic algebra skills.
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- ↑ http://www.mathsteacher.com.au/year9/ch05_simult/01_sub/method.htm
- ↑ http://www.purplemath.com/modules/systlin4.htm
- ↑ https://flexbooks.ck12.org/cbook/ck-12-cbse-math-class-10/section/3.5/primary/lesson/solving-simultaneous-linear-equations-by-substitution/
- ↑ https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-systems-topic/cc-8th-systems-with-substitution/a/systems-of-equations-with-substitution
- ↑ https://www.bbc.co.uk/bitesize/guides/zd9dmp3/revision/2
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Simultaneous Equations
Simultaneous equations are two or more algebraic equations that share common variables and are solved at the same time (that is, simultaneously). For example, equations x + y = 5 and x - y = 6 are simultaneous equations as they have the same unknown variables x and y and are solved simultaneously to determine the value of the variables. We can solve simultaneous equations using different methods such as substitution method, elimination method, and graphically.
In this article, we will explore the concept of simultaneous equations and learn how to solve them using different methods of solving. We shall discuss the simultaneous equations rules and also solve a few examples based on the concept for a better understanding.
What are Simultaneous Equations?
Simultaneous equations are two or more algebraic equations with the same unknown variables and the same value of the variables satisfies all such equations. This implies that the simultaneous equations have a common solution. Some of the examples of simultaneous equations are:
- 2x - 4y = 4, 5x + 8y = 3
- 2a - 3b + c = 9, a + b + c = 2, a - b - c = 9
- 3x - y = 5, x - y = 4
- a 2 + b 2 = 9, a 2 - b 2 = 16
We can solve such a set of equations using different methods. Let us discuss different methods to solve simultaneous equations in the next section.
Solving Simultaneous Equations
We use different methods to solve simultaneous equations. Some of the common methods are:
- Substitution Method
- Elimination Method
- Graphical Method
Simultaneous equations can have no solution, an infinite number of solutions, or unique solutions depending upon the coefficients of the variables. We can also use the method of cross multiplication and determinant method to solve linear simultaneous equations in two variables . We can add/subtract the equations depending upon the sign of the coefficients of the variables to solve them.
To solve simultaneous equations, we need the same number of equations as the number of unknown variables involved. We shall discuss each of these methods in detail in the upcoming sections with examples to understand their applications properly.
Simultaneous Equations Rules
To solve simultaneous equations, we follow certain rules first to simplify the equations. Some of the important rules are:
- Simplify each side of the equation first by removing the parentheses, if any.
- Combine the like terms .
- Isolate the variable terms on one side of the equation.
- Then, use the appropriate method to solve for the variable.
Solving Simultaneous Equations Using Substitution Method
Now that we have discussed different methods to solve simultaneous equations. Let us solve a few examples using the substitution method to understand it better. Consider a system of equations x + y = 4 and 2x - 3y = 9. Now, we will find the value of one variable in terms of another variable using one of the equations and substitute it into the other equation. We have
x + y = 4 --- (1)
2x - 3y = 9 --- (2)
From (1), we have
x = 4 - y --- (3)
Substituting this in (2), we get
2(4 - y) - 3y = 9
⇒ 8 - 2y - 3y = 9
⇒ 8 - 5y = 9
Isolating the variable term to one side of the equation, we have
⇒ -5y = 9 - 8
⇒ y = 1/(-5)
Substituting the value of y in (3), we have
x = 4 - (-1/5)
= (20 + 1)/5
Answer: So, the solution of the simultaneous equations x + y = 4 and 2x - 3y = 9 is x = 21/5 and y = -1/5.
Solving Simultaneous Equations By Elimination Method
To solve simultaneous equations by the elimination method, we eliminate a variable from one equation using another to find the value of the other variable. Let us solve an example to understand find the solution of simultaneous equations using the elimination method. Consider equations 2x - 5y = 3 and 3x - 2y = 5. We have
2x - 5y = 3 --- (1)
and 3x - 2y = 5 --- (2)
Here, we will eliminate the variable y, so we find the LCM of the coefficients of y. LCM (5, 2) = 10. So, multiply equation (1) by 2 and equation (2) by 5. So, we have
[ 2x - 5y = 3 ] × 2
⇒ 4x - 10y = 6 --- (3)
[ 3x - 2y = 5 ] × 5
⇒ 15x - 10y = 25 --- (4)
Now, subtracting equation (3) from (4), we have
(15x - 10y) - (4x - 10y) = 25 - 6
⇒ 15x - 10y - 4x + 10y = 19
⇒ (15x - 4x) + (-10y + 10y) = 19
⇒ 11x + 0 = 19
⇒ x = 19/11
Now, substituting this value of x in (1), we have
2(19/11) - 5y = 3
⇒ 38/11 - 5y = 3
⇒ 5y = 38/11 - 3
⇒ 5y = (38 - 33) / 11
⇒ y = 5/(11×5)
So, the solution of the simultaneous equations 2x - 5y = 3 and 3x - 2y = 5 using the elimination method is x = 19/11 and y = 1/11.
Solving Simultaneous Equations Graphically
In this section, we will learn to solve the simultaneous equations using the graphical method. We will plot the lines on the coordinate plane and then find the point of intersection of the lines to find the solution. Consider simultaneous equations x + y = 10 and x - y = 4. Now, find two points (x, y) satisfying for each equation such that the equation holds.
For x + y = 10, we have
So, we have coordinates (0, 10) and (10, 0). Plot them and join the points and plot the line x + y = 10.
For equation x - y = 4, we have
So, we have coordinates (0, -4) and (4, 0). Plot them and join the points and plot the line x - y = 4.
Now, as we have plotted the two lines, find their intersecting point. The two lines x + y = 10 and x - y = 4 intersect each other at (7, 3). So, we have found the solution of the simultaneous equations x + y = 10 and x - y = 4 graphically which is x = 7 and y = 3.
Important Notes on Simultaneous Equations
- Simultaneous equations are two or more algebraic equations that share common variables and are solved at the same time.
- Simultaneous equations can be solved using different methods such as substitution method, elimination method, and graphically.
- We can also use the cross multiplication and determinant method to solve simultaneous linear equations in two variables.
☛ Related Articles:
- Solutions of a Linear Equation
- Simultaneous Linear Equations
Simultaneous Equations Examples
Example 1: Solve the simultaneous equations 2x - y = 5 and y - 4x = 1 using the appropriate method.
Solution: To solve 2x - y = 5 and y - 4x = 1, we will use the elimination method as it is easy to eliminate the variable y by adding the two equations. So, we have
2x - y = 5 --- (1)
y - 4x = 1 --- (2)
Adding (1) and (2), we get
(2x - y) + (y - 4x) = 5 + 1
⇒ 2x - y + y - 4x = 6
⇒ -2x = 6
⇒ x = -6/2
Substitute this value of x in (1)
2(-3) - y = 5
⇒ -6 - y = 5
⇒ y = -6 - 5
Answer: Solution of simultaneous equations 2x - y = 5 and y - 4x = 1 is x = -3 and y = -11.
Example 2: Find the solution of the simultaneous equations 2x - 4y + z = 2, x + 5y - 3z = 7, 3x + 2y - z = 10 using the substitution method.
Solution: We have
2x - 4y + z = 2 --- (1)
x + 5y - 3z = 7 --- (2)
3x + 2y - z = 10 --- (3)
z = 2 - 2x + 4y
Substituting this value of z in (2) and (3),
x + 5y - 3(2 - 2x + 4y) = 7
⇒ x + 5y - 6 + 6x - 12y = 7
⇒ 7x - 7y = 13 --- (4)
3x + 2y - (2 - 2x + 4y) = 10
3x + 2y - 2 + 2x - 4y = 10
⇒ 5x - 2y = 12 --- (5)
Now, solving the two-variable equations (4) and (5), multiply (4) by 2 and (5) by 7, we have
[7x - 7y = 13 ] × 2 and [5x - 2y = 12 ] × 7
⇒ 14x - 14y = 26 and 35x - 14y = 84
Now, subtracting the above two equations, we have
(14x - 14y) - (35x - 14y)= 26 - 84
⇒ 14x - 35x - 14y + 14y = -58
⇒ -21x = -58
⇒ x = 58/21 --- (A)
Substitute the value of x in (5)
5(58/21) - 2y = 12
⇒ 290/21 - 2y = 12
⇒ 2y = 290/21 - 12
= (290 - 252)/21
⇒ y = 19/21 --- (B)
Substituting the values of x and y in z = 2 - 2x + 4y, we have
z = 2 - 2(58/21) + 4(19/21)
= (42 - 116 + 76)/21
= 2/21 --- (C)
From (A), (B), (C), we have x = 58/21, y = 19/21, and z = 2/21
Answer: Solution is x = 58/21, y = 19/21, and z = 2/21.
Example 3: Find the solution of simultaneous equations x - y = 10 and 2x + y = 9.
Solution: We will solve the given equations using the elimination method.
Adding x - y = 10 and 2x + y = 9, we have
(x - y) + (2x + y) = 10 + 9
⇒ x + 2x - y + y = 19
⇒ 3x = 19
⇒ x = 19/3
So, we have
19/3 - y = 10
⇒ y = 19/3 - 10
= (19 - 30)/3
Answer: The solution of x - y = 10 and 2x + y = 9 is x = 19/3 and y = -11/3.
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Simultaneous Equations Questions
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FAQs on Simultaneous Equations
Simultaneous equations are two or more algebraic equations that share common variables and are solved at the same time (that is, simultaneously).
How to Solve Simultaneous Equations?
What is the substitution method in simultaneous equations.
According to the substitution method, we obtain the value of one variable in terms of another and then substitute that into another equation to find the value of the other variable.
What is the Rule for Simultaneous Equations?
Some of the important rules of simultaneous equations are:
- Combine the like terms.
What are Linear Simultaneous Equations?
Linear simultaneous equations refer to simultaneous equations where the degree of the variables is one.
How to Solve 3 Simultaneous Equations?
We can solve 3 simultaneous equations using various methods such as:
It also depends upon the number of variables involved.
What are the Three Methods to Solve Simultaneous Equations?
The three methods to solve simultaneous equations are:
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Simultaneous Equations – GCSE Maths Revision Guide
Simultaneous equations are a key part of algebra for GCSE maths. To do well with simultaneous equations you need to be confident solving difficult linear equations. Check out our linear equations guide and worksheet if you need to revise this before going any further. Once you’re confident with linear equations read on and find out how to solve simultaneous equations.
Up to this point you have learnt about solving one equation with one unknown (represented by a letter). To solve the equation you find the value of the unknown. Simultaneous equations allow you to find two unknowns, as long as you have two equations and both have the same two unknowns. The two equations are solved simultaneously (at the same time) to find the values of both unknowns.
You need to learn two methods to solve simultaneous equations for GCSE maths: the elimination method and the substitution method. I have set out guides to both methods below, including some examples. I’ve written in key parts of working that you need to show the examiner. You will use the elimination method most often, so let’s start there. You can also find more great revision guides and questions on our free resources page .
Simultaneous Equations – Elimination Method
With this method you will need to combine the two equations by either adding or subtracting one from the other. In doing so you need to eliminate one of the unknowns. You then solve the remaining equation to find one unknown and substitute your answer back into one of the original equations to find the second unknown. Take a look at the worked example below.
Worked Example 1
Firstly, label the equations number one and two to help with your thinking and to help the examiner see your working.
Next, combine the equations and eliminate the x (we eliminate x rather than y here because the coefficients of x are the same) by subtracting equation one from equation two. Make sure you subtract everything, including the x, y and numbers after the equals sign.
We now know the value of y. Substitute y = 1 into one of the original equations. I’ve used equation one here:
Then solve this final equation to find the value of x.
That’s it. We’ve solved the simultaneous equations to find y = 1 and x = 2.
Worked Example 2
Label the equations one and two:
This time we will eliminate the y because the coefficients of y are already the same. Notice that equation one includes 2y and equation two includes -2y. When the terms you want to eliminate have different signs, you should add the two equations together. This will ensure that you eliminate the y.
Next, substitute x = 3 into equation one.
Do you need any further help before continuing? Check out how our online maths tutors can help you reach the next level with your GCSE revision.
Worked Example 3
Notice that this pair of simultaneous equations do not have an unknown with the same coefficient (as the earlier examples did). When this is the case, you will need to do an extra step at the beginning. After labelling the equations, you need to find the lowest common multiple of the coefficients for x or for y. Multiply up to this lowest common multiple to ensure the coefficients of one of the unknowns are the same. Then follow the same process as with the earlier examples to combine the equations and solve.
Substitute y = 2 into equation one:
Simultaneous Equations – Substitution Method
The second method you need to know to solve some simultaneous equations for GCSE maths is the substitution method. As with the elimination method: we will combine the two equations, solve for one unknown and then use that answer to find the second unknown. However, with this method we combine the equations initially by substituting part of one equation into the other (hence the name: substitution method).
Worked Example 4
Start by labelling the equations one and two. Then substitute equation two into equation one. In equation two y is the subject. This allows us to substitute (x – 8) in for y.
Finally, substitute x = 6 into equation two and solve for y.
Keep Practicing
That is everything you need to know about simultaneous equations for GCSE maths. Make sure you practice both the elimination and substitution methods once you have read through the information above a few times. You can find plenty of practice simultaneous questions questions on our worksheet page. It is only by doing lots of questions yourself that you will properly learn and understand the topic. Follow the links below to try some past paper questions as well. Let us know if you have any questions as you continue revising. You can contact us directly through our website to ask any questions or book an expert maths tutor for a personalised lesson.
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- Math Article
Simultaneous Equations
Before starting with simultaneous equations, let’s recall what are equations in maths and the types of equations. In Mathematics, an equation is a mathematical statement in which two things should be equal to each other. An equation consists of two expressions on each side of an equal sign (=). It consists of two or more variables. In short, the L.H.S value should be equal to the R.H.S value. While substituting the values of the variables in an equation, it should prove its equality. There are different types of equations in Maths, such as:
- Linear Equation
- Quadratic Equation
- Polynomial Equations
and so on. In this article, we are going to discuss the simultaneous equations which involve two variables along with different methods to solve.
What are Simultaneous Equations?
The simultaneous equation is an equation that involves two or more quantities that are related using two or more equations. It includes a set of few independent equations. The simultaneous equations are also known as the system of equations, in which it consists of a finite set of equations for which the common solution is sought. To solve the equations, we need to find the values of the variables included in these equations.
The system of equations or simultaneous equations can be classified as:
- Simultaneous linear equations (Or) System of linear equations
- Simultaneous non-linear equations
- System of bilinear equations
- Simultaneous polynomial equations
- System of differential equations
Here, you will learn the methods of solving simultaneous linear equations along with examples.
The general form of simultaneous linear equations is given as:
dx + ey = f
Methods for Solving Simultaneous Equations
The simultaneous linear equations can be solved using various methods. There are three different approaches to solve the simultaneous equations such as substitution, elimination, and augmented matrix method. Among these three methods, the two simplest methods will effectively solve the simultaneous equations to get accurate solutions. Here we are going to discuss these two important methods, namely,
- Elimination Method
- Substitution Method
Apart from those methods, we can also the system of linear equations using Cramer’s rule .
If the simultaneous linear equations contain only two variables, we may also use the cross-multiplication method to find their solution.
Simultaneous Equation Example
Let us now understand how to solve simultaneous equations through the above-mentioned methods. We will get the value of a and b to find the solution for the same. x and y are the two variables in these equations. Go through the following problems which use substitution and elimination methods to solve the simultaneous equations.
Try Out: Simultaneous Equation Solver
Solving Simultaneous Linear Equations Using Elimination Method
Go through the solved example given below to understand the method of solving simultaneous equations by the elimination method along with steps.
Example: Solve the following simultaneous equations using the elimination method.
4a + 5b = 12,
3a – 5b = 9
The two given equations are
4a + 5b = 12 …….(1)
3a – 5b = 9……….(2)
Step 1: The coefficient of variable ’b’ is equal and has the opposite sign to the other equation. Add equations 1 and 2 to eliminate the variable ‘b’.
Step 2: The like terms will be added.
(4a+3a) +(5b – 5b) = 12 + 9
Step 3: Bring the coefficient of a to the R.H.S of the equation
Step 4: Dividing the R.H. S of the equation, we get a = 3
Step 5: Now, substitute the value a=3 in the equation (1), it becomes
4(3) + 5b = 12,
12 + 5b = 12
b = 0/5 = 0
Step 6: Hence, the solution for the given simultaneous equations is a = 3 and b = 0.
Solving Simultaneous Linear Equations Using Substitution Method
Below is the solved example with steps to understand the solution of simultaneous linear equations using the substitution method in a better way.
Example: Solve the following simultaneous equations using the substitution method.
b = a + 2 ————–(1)
a + b = 4 ————–(2)
We will solve it step-wise:
Step 1: Substitute the value of b into the second equation. We will get,
a + (a + 2) = 4
Step 2: Solve for a
a +a + 2 = 4
2a = 4 – 2
a = 2/2 = 1
Step 3: Substitute this value of a in equation 1
step 4: Hence, the solution for the given simultaneous equations is: a = 1 and b = 3
Practice Problems
- Solve: 5x + 3y = 7 and -3x + 5y = 23
- Solve for a and b: 10a – 8b = 6 10a – 9b = -2
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Simultaneous Equations
Here is everything you need to know about simultaneous equations for GCSE maths (Edexcel, AQA and OCR).
You’ll learn what simultaneous equations are and how to solve them algebraically. We will also discuss their relationship to graphs and how they can be solved graphically.
Look out for the simultaneous equations worksheets and exam questions at the end.
What are simultaneous equations?
Simultaneous equations are two or more algebraic equations that share variables e.g. x and y .
They are called simultaneous equations because the equations are solved at the same time.
For example, below are some simultaneous equations:
Each of these equations on their own could have infinite possible solutions.
However when we have at least as many equations as variables we may be able to solve them using methods for solving simultaneous equations.
We can consider each equation as a function which, when displayed graphically, may intersect at a specific point. This point of intersection gives the solution to the simultaneous equations.
When we draw the graphs of these two equations, we can see that they intersect at (1,5).
So the solutions to the simultaneous equations in this instance are:
x = 1 and y = 5
Solving simultaneous equations
When solving simultaneous equations you will need different methods depending on what sort of simultaneous equations you are dealing with. There are two sorts of simultaneous equations you will need to solve:
- linear simultaneous equations
- quadratic simultaneous equations
A linear equation contains terms that are raised to a power that is no higher than one.
Linear simultaneous equations are usually solved by what’s called the elimination method (although the substitution method is also an option for you ) .
Solving simultaneous equations using the elimination method requires you to first eliminate one of the variables, next find the value of one variable, then find the value of the remaining variable via substitution. Examples of this method are given below.
A quadratic equation contains terms that are raised to a power that is no higher than two.
Quadratic simultaneous equations are solved by the substitution method.
See also: 15 Simultaneous equations questions
Simultaneous equations worksheets
Get your free simultaneous equations worksheet of 20+ questions and answers. Includes reasoning and applied questions.
How to solve simultaneous equations
To solve pairs of simultaneous equations you need to:
- Use the elimination method to get rid of one of the variables.
- Find the value of one variable.
- Find the value of the remaining variables using substitution.
- Clearly state the final answer.
- Check your answer by substituting both values into either of the original equations.
How do you solve pairs of simultaneous equations?
See the examples below for how to solve the simultaneous linear equations using the three most common forms of simultaneous equations.
See also: Substitution
Quadratic simultaneous equations
Quadratic simultaneous equations have two or more equations that share variables that are raised to powers up to 2 e.g. x^{2} and y^{2} . Solving quadratic simultaneous equations algebraically by substitution is covered, with examples, in a separate lesson.
Step-by-step guide: Quadratic simultaneous equations
Simultaneous equations examples
For each of the simultaneous equations examples below we have included a graphical representation.
Step-by-step guide : Solving simultaneous equations graphically
Example 1: Solving simultaneous equations by elimination (addition)
- Eliminate one of the variables.
By adding the two equations together we can eliminate the variable y .
2 Find the value of one variable.
3 Find the value of the remaining variable via substitution.
We know x = 3 so we can substitute this value into either of our original equations.
4 Clearly state the final answer.
5 Check your answer by substituting both values into either of the original equations.
This is correct so we can be confident our answer is correct.
Graphical representation of solving by elimination (addition)
When we draw the graphs of these linear equations they produce two straight lines. These two lines intersect at (1,5). So the solution to the simultaneous equations is x = 3 and y = 2 .
Example 2: Solving simultaneous equations by elimination (subtraction)
By subtracting the two equations we can eliminate the variable b .
NOTE: b − b = 0 so b is eliminated
3 Find the value of the remaining variable/s via substitution.
We know a = 2 so we can substitute this value into either of our original equations.
Graphical representation of solving by elimination (subtraction)
When graphed these two equations intersect at (1,5). So the solution to the simultaneous equations is a = 2 and b = 6 .
Example 3: Solving simultaneous equations by elimination (different coefficients)
Notice that adding or subtracting the equations does not eliminate either variable (see below).
This is because neither of the coefficients of h or i are the same. If you look at the first two examples this was the case.
So our first step in eliminating one of the variables is to make either coefficients of h or i the same.
We are going to equate the variable of h .
Multiply every term in the first equation by 2 .
Multiply every term in the second equation by 3 .
Now the coefficients of h are the same in each of these new equations, we can proceed with our steps from the first two examples. In this example, we are going to subtract the equations.
Note: 6h − 6h = 0 so h is eliminated
Careful : 16 − − 6 = 22
We know i = − 2 so we can substitute this value into either of our original equations.
Graphical representation of solving by elimination (different coefficients)
When graphed these two equations intersect at (1,5). So the solution to the simultaneous equations is h = 4 and i = − 2 .
Example 4: Worded simultaneous equation
David buys 10 apples and 6 bananas in a shop. They cost £5 in total. In the same shop, Ellie buys 3 apples and 1 banana. She spends £1.30 in total. Find the cost of one apple and one banana.
Additional step: conversion
We need to convert this worded example into mathematical language. We can do this by representing apples with a and bananas with b .
Notice we now have equations where we do not have equal coefficients (see example 3).
We are going to equate the variable of b .
Multiply every term in the first equation by 1 .
Multiply every term in the second equation by 6 .
Now the coefficients of b are the same in each equation we can proceed with our steps from the previous examples. In this example, we are going to subtract the equations.
NOTE: 6b − 6b = 0 so b is eliminated
16 − − 6 = 22
Note : we ÷ (− 8) not 8
We know a = 0.35 so we can substitute this value into either of our original equations.
1 apple costs £0.35 (or 35p ) and 1 banana costs £0.25 (or 25p ).
Graphical representation of the worded simultaneous equatio
When graphed these two equations intersect at (1,5). So the solution to the simultaneous equations is a = 0.35 and b = 0.25 .
Common misconceptions
- Incorrectly eliminating a variable. Using addition to eliminate one variable when you should subtract (and vice-versa).
- Errors with negative numbers. Making small mistakes when +, −, ✕, ÷ with negative numbers can lead to an incorrect answer. Working out the calculation separately can help to minimise error. Step by step guide: Negative numbers (coming soon)
- Not multiplying every term in the equation. Mistakes when multiplying an equation. For example, forgetting to multiply every term by the same number.
- Not checking the answer using substitution. Errors can quickly be spotted by substituting your solutions in the original first or second equations to check they work.
Practice simultaneous equations questions
1. Solve the Simultaneous Equation
6x +3y = 48 6x +y =26
Subtracting the second equation from the first equation leads to a single variable equation. Use this equation to determine the value of y , then substitute this value into either equation to determine the value of x .
2. Solve the Simultaneous Equation x -2y = 8 x -3y =3
Subtracting the second equation from the first equation leads to a single variable equation, which determines the value of y . Substitute this value into either equation to determine the value of x .
3. Solve the Simultaneous Equation 4x +2y = 34 3x +y =21
In this case, a good strategy is to multiply the second equation by 2 . We can then subtract the first equation from the second to leave an equation with a single variable. Once this value is determined, we can substitute it into either equation to find the value of the other variable.
4. Solve the Simultaneous Equation:
15x -4y = 82 5x -9y =12
In this case, a good strategy is to multiply the second equation by 3 . We can then subtract the second equation from the first to leave an equation with a single variable. Once this value is determined, we can substitute it into either equation to find the value of the other variable.
Simultaneous equations GCSE questions
1. Solve the simultaneous equations
\begin{array}{l} 5x=-10 \\ x=-2 \end{array} or correct attempt to find y
One unknown substituted back into either equation
2. Solve the simultaneous equations
Correct attempt to multiple either equation to equate coefficients e.g.
Correct attempt to find y or x ( 16y=56 or 16x = 24 seen)
3. Solve the simultaneous equations
Correct attempt to find y or x ( 13x=91 or 13y=-39 seen)
Learning checklist
- Solve two simultaneous equations with two variables (linear/linear) algebraically
- Derive two simultaneous equations, solve the equation(s) and interpret the solution
The next lessons are
- Maths formulas
- Types of graphs
- Interpreting graphs
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Solving simultaneous equations
Simultaneous means 'at the same time'. When solving simultaneous equations you are trying to find the values of the unknowns that satisfy both equations at the same time.
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Simultaneous equations with one linear and one non-linear - Higher
\[x + 2y = -3\]
\[xy = -14\]
Again, algebraic skills of substitution and factorising are required to solve these equations.
Rewriting the first equation gives \(x = -3 – 2y\)
This can be substituted into the second equation to give an equation that can be factorised and solved.
\[(-3 – 2y)y = -14\]
\[-3y – 2y^2 = -14\]
\[0 = 2y^2 + 3y -14\]
Factorise this equation:
\[(2y + 7)(y – 2) = 0\]
If the product of two numbers is zero, then one or both numbers must also be equal to zero. To solve, put each bracket equal to zero.
\[2y + 7 = 0\]
\[2y = -7\]
\[y = -3.5\]
\[y – 2 = 0\]
To find the values for \(x\) substitute the two values for \(y\) into the equation \(x = -3 – 2y\)
When \(y = -3.5\) ,
\[x = -3 + 7\]
When \(y = 2\) ,
\[x = -3 -4\]
The answers are now in pairs: when \(x = 4, y = -3.5\) and when \(x = -7, y = 2\) .
Solving linear and quadratic equations graphically - Higher
Simultaneous equations that contain a quadratic equation can also be solved graphically. As with solving algebraically, there will usually be two pairs of solutions.
Solve the simultaneous equations \(y = x^2\) and \(y = x + 2\) .
\[y = x^2\]
\[y = x + 2\]
Plot the graphs on the axes and look for the points of intersection.
The two points of intersection are at (2, 4) and (-1, 1) so \(x = 2\) and \(y = 4\) , and \(x = -1\) and \(y = 1\) .
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Maths revision video and notes on the topic solving simultaneous equations. GCSE Revision. GCSE Papers . Edexcel Exam Papers OCR Exam Papers AQA Exam Papers. ... Maths Genie Limited is a company registered in England and Wales with company number 14341280. Registered Office: 143 Lynwood, Folkestone, Kent, CT19 5DF. ...
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Take that value of x, and substitute it into the first equation given above (x + y = 3). With that substitution the first equation becomes (1+y) + y = 3. That means 1 + 2y = 3. Subtract 1 from each side: 2y = 2. So y = 1. Substitute that value of y into either of the two original equations, and you'll get x = 2.
This video goes through how to solve a linear and quadratic simultaneous equation using the substitution method and solving using factorising. This should help for anyone trying to get an...
The most common method for solving simultaneous equations is the elimination method which means one of the unknowns will be removed from each equation. The remaining unknown can then be...
Simultaneous Equations Examples. Example 1: Solve the simultaneous equations 2x - y = 5 and y - 4x = 1 using the appropriate method. Solution: To solve 2x - y = 5 and y - 4x = 1, we will use the elimination method as it is easy to eliminate the variable y by adding the two equations. So, we have.
PDF Name: GCSE (1 - 9) Quadratic Simultaneous Equations - Maths Genie. Quadratic Simultaneous Equations Name: _____ Instructions • Use black ink or ball-point pen. • Answer all questions. • Answer the questions in the spaces provided - there may be more space than you need. • Diagrams are NOT accurately drawn, unless otherwise indicated.
Make sure you subtract everything, including the x, y and numbers after the equals sign. We now know the value of y. Substitute y = 1 into one of the original equations. I've used equation one here: Then solve this final equation to find the value of x. That's it. We've solved the simultaneous equations to find y = 1 and x = 2.
Solve this pair of simultaneous equations graphically: y = 2x +1 y = 4x +3 y = 2 x + 1 y = 4 x + 3. Identify if the equations are linear or quadratic. Both the equations are linear. This means you will be drawing two straight lines which will intersect at one point only. 2 Draw each equation on the same set o f axes.
Step 1: The coefficient of variable 'b' is equal and has the opposite sign to the other equation. Add equations 1 and 2 to eliminate the variable 'b'. Step 2: The like terms will be added. (4a+3a) + (5b - 5b) = 12 + 9. 7a = 21. Step 3: Bring the coefficient of a to the R.H.S of the equation. a = 21/ 7.
Example 2: Solving simultaneous equations by elimination (subtraction) Solve: Eliminate one of the variables. By subtracting the two equations we can eliminate the variable b. NOTE: b − b = 0 so b is eliminated. 2 Find the value of one variable. 3 Find the value of the remaining variable/s via substitution.
Simultaneous equations that contain a quadratic equation can also be solved graphically. As with solving algebraically, there will usually be two pairs of solutions. Example. Solve the ...