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## How to Solve Simultaneous Equations Using Substitution Method

Last Updated: October 10, 2022 References

## Finding the Value of y

## Finding the Value of x

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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

## What are Simultaneous Equations?

- 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

## Solving Simultaneous Equations

We use different methods to solve simultaneous equations. Some of the common methods are:

## Simultaneous Equations Rules

- 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

Substituting this in (2), we get

Isolating the variable term to one side of the equation, we have

Substituting the value of y in (3), we have

## Solving Simultaneous Equations By Elimination Method

Now, subtracting equation (3) from (4), we have

(15x - 10y) - (4x - 10y) = 25 - 6

⇒ (15x - 4x) + (-10y + 10y) = 19

Now, substituting this value of x in (1), we have

## Solving Simultaneous Equations Graphically

For equation x - y = 4, we have

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.

## Simultaneous Equations Examples

Example 1: Solve the simultaneous equations 2x - y = 5 and y - 4x = 1 using the appropriate method.

Substitute this value of x in (1)

Answer: Solution of simultaneous equations 2x - y = 5 and y - 4x = 1 is x = -3 and y = -11.

Substituting this value of z in (2) and (3),

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

Substitute the value of x in (5)

Substituting the values of x and y in z = 2 - 2x + 4y, we have

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

Answer: The solution of x - y = 10 and 2x + y = 9 is x = 19/3 and y = -11/3.

go to slide go to slide go to slide

## Simultaneous Equations Questions

## FAQs on Simultaneous Equations

## How to Solve Simultaneous Equations?

What is the substitution method in simultaneous equations.

## What is the Rule for Simultaneous Equations?

Some of the important rules of simultaneous equations are:

## What are Linear Simultaneous Equations?

## 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:

## Download Maths Genie Quadratic Simultaneous Equations Worksheet Answers:

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## Worked Example 1

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:

Next, substitute x = 3 into equation one.

## Worked Example 3

Substitute y = 2 into equation one:

## Simultaneous Equations – Substitution Method

## Worked Example 4

Finally, substitute x = 6 into equation two and solve for y.

## Keep Practicing

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## Simultaneous Equations

## What are Simultaneous 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:

## Methods for Solving Simultaneous Equations

Apart from those methods, we can also the system of linear equations using Cramer’s rule .

## Simultaneous Equation Example

Try Out: Simultaneous Equation Solver

## Solving Simultaneous Linear Equations Using Elimination Method

Example: Solve the following simultaneous equations using the elimination method.

Step 2: The like terms will be added.

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

Step 6: Hence, the solution for the given simultaneous equations is a = 3 and b = 0.

## Solving Simultaneous Linear Equations Using Substitution Method

Example: Solve the following simultaneous equations using the substitution method.

Step 1: Substitute the value of b into the second equation. We will get,

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

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## Simultaneous Equations

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.

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:

## Solving simultaneous equations

A linear equation contains terms that are raised to a power that is no higher than one.

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

## 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?

## Quadratic simultaneous equations

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)

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)

## 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)

## Example 3: Solving simultaneous equations by elimination (different coefficients)

Notice that adding or subtracting the equations does not eliminate either variable (see below).

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 .

Note: 6h − 6h = 0 so h is eliminated

We know i = − 2 so we can substitute this value into either of our original equations.

## Graphical representation of solving by elimination (different coefficients)

## Example 4: Worded simultaneous equation

## Additional step: conversion

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 .

NOTE: 6b − 6b = 0 so b is eliminated

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

## 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

2. Solve the Simultaneous Equation x -2y = 8 x -3y =3

3. Solve the Simultaneous Equation 4x +2y = 34 3x +y =21

4. Solve the Simultaneous Equation:

## 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

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## Solving simultaneous equations

## Simultaneous equations with one linear and one non-linear - Higher

Again, algebraic skills of substitution and factorising are required to solve these equations.

Rewriting the first equation gives \(x = -3 – 2y\)

To find the values for \(x\) substitute the two values for \(y\) into the equation \(x = -3 – 2y\)

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

Solve the simultaneous equations \(y = x^2\) and \(y = x + 2\) .

Plot the graphs on the axes and look for the points of intersection.

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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 ...