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## Unit 6: Lesson 3

- Systems of equations with elimination: King's cupcakes
- Elimination strategies
- Systems of equations with elimination: x-4y=-18 & -x+3y=11
- Systems of equations with elimination: potato chips

## Systems of equations with elimination (and manipulation)

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## Video transcript

## Elimination by Multiplication - Examples & Practice - Expii

## Module 16: Linear Systems

The elimination method with multiplication, learning outcomes.

- Use the elimination method with multiplication
- Express the solution of a dependent system of equations containing two variables

## Solve a system of equations when multiplication is necessary to eliminate a variable

[latex]\begin{array}{r}3x+4y=52\\5x+y=30\end{array}[/latex]

The following example takes you through all the steps to find a solution to this system.

Solve for [latex]x[/latex] and [latex]y[/latex].

Equation A: [latex]3x+4y=52[/latex]

Equation B: [latex]5x+y=30[/latex]

Multiply the second equation by [latex]−4[/latex] so they do have the same coefficient.

Rewrite the system and add the equations.

[latex]\begin{array}{r}3x+4y=52\,\,\,\,\,\,\,\\−20x–4y=−120\end{array}[/latex]

[latex]\begin{array}{l}−17x=-68\\\,\,\,\,\,\,\,\,\,\,x=4\end{array}[/latex]

Substitute [latex]x=4[/latex] into one of the original equations to find y.

[latex]\begin{array}{r}3x+4y=52\\3\left(4\right)+4y=52\\12+4y=52\\4y=40\\y=10\end{array}[/latex]

The solution is [latex](4, 10)[/latex].

Solve the given system of equations by the elimination method.

[latex]\begin{array}{l}3x+5y=-11\hfill \\ x - 2y=11\hfill \end{array}[/latex]

Now multiply the bottom equation by [latex]−3[/latex].

Next add the equations, and solve for [latex]y[/latex].

[latex]\begin{array}{r}15x+20y=260\\−15x–3y=\,–90\\17y=170\\y=\,\,\,10\end{array}[/latex]

Substitute [latex]y=10[/latex] into one of the original equations to find [latex]x[/latex].

You arrive at the same solution as before.

Solve the given system of equations in two variables by elimination.

Then, we add the two equations together.

Substitute [latex]y=-4[/latex] into the original first equation.

[latex]\begin{array}{c}2x+3\left(-4\right)=-16\\ 2x - 12=-16\\ 2x=-4\\ x=-2\end{array}[/latex]

The solution is [latex]\left(-2,-4\right)[/latex]. Check it in the second original equation.

## How To: Given a system of equations, solve using the elimination method

- Write both equations with x and y -variables on the left side of the equal sign and constants on the right.
- Write one equation above the other, lining up corresponding variables. If one of the variables in the top equation has the opposite coefficient of the same variable in the bottom equation, add the equations together, eliminating one variable. If not, use multiplication by a nonzero number so that one of the variables in the top equation has the opposite coefficient of the same variable in the bottom equation, then add the equations to eliminate the variable.
- Solve the resulting equation for the remaining variable.
- Substitute that value into one of the original equations and solve for the second variable.
- Check the solution by substituting the values into the other equation.

Now multiply the second equation by [latex]-1[/latex] so that we can eliminate the x -variable.

Add the two equations to eliminate the x -variable and solve the resulting equation.

Substitute [latex]y=7[/latex] into the first equation.

The solution is [latex]\left(\dfrac{11}{2},7\right)[/latex]. Check it in the other equation.

[latex]\begin{array}{c}2x-y=4\\ 2(\dfrac{11}{2})-7=4\\ 11-7=4 \\ 4=4\end{array}[/latex]

Find a solution to the system of equations using the elimination method .

[latex]\begin{array}{c}x+3y=2\\ 3x+9y=6\end{array}[/latex]

We can see that there will be an infinite number of solutions that satisfy both equations.

## Contribute!

- Revision and Adaptation. Provided by : Lumen Learning. License : CC BY: Attribution
- Ex: System of Equations Using Elimination (Infinite Solutions). Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/NRxh9Q16Ulk . License : CC BY: Attribution
- Unit 14: Systems of Equations and Inequalities, from Developmental Math: An Open Program. Provided by : Monterey Institute of Technology and Education. Located at : . License : CC BY: Attribution
- Ex 2: Solve a System of Equations Using the Elimination Method. Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/_liDhKops2w . License : CC BY: Attribution

## How Do You Solve a System of Equations Using the Elimination by Multiplication Method?

- linear equations
- system of equations
- 2 equations
- find intersection
- solve system of equations
- solve by elimination
- elimination
- variable elimination
- add equations
- multiply equation
- multiply by negative
- independent
- 2 variables

## Background Tutorials

## What's a System of Linear Equations?

## Evaluating Expressions

## What is a Variable?

## Further Exploration

Solving systems using elimination.

## How Do You Solve a System of Equations Using the Elimination by Addition Method?

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## Elimination Using Multiplication (Jump to: Lecture | Video )

Below, we have a system of equations:

We multiply the first equation by "2" to prepare it for elimination by subtraction.

Now that we know "x", we can use this information to solve for "y".

· Solve a system of equations when no multiplication is necessary to eliminate a variable.

· Solve a system of equations when multiplication is necessary to eliminate a variable.

· Recognize systems that have no solution or an infinite number of solutions.

· Solve application problems using the elimination method.

Using Addition to Eliminate a Variable

If you add the two equations, x – y = −6 and x + y = 8 together, as noted above, watch what happens.

Let’s see how this system is solved using the elimination method.

2 x + y =12 → 2 x + y = 12 → 2 x + y = 12

− 3 x + y = 2 → − ( − 3 x + y ) = − (2) → 3 x – y = − 2

You have eliminated the y variable, and the problem can now be solved. See the example below.

Substitute y = 3 into one of the original equations.

Change one of the equations to its opposite, add and solve for x .

Substitute x = 2 into one of the original equations and solve for y .

Using Multiplication and Addition to Eliminate a Variables

3 x + 4 y = 52 → 3 x + 4 y = 52 → 3 x + 4 y = 52

5 x + y = 30 → − 4(5 x + y ) = − 4(30) → − 20 x – 4 y = − 120

Multiply the second equation by − 4 so they do have the same coefficient.

Rewrite the system, and add the equations.

Substitute x = 4 into one of the original equations to find y .

Let’s remove the variable x this time. Multiply Equation A by 5 and Equation B by − 3.

Now multiply the bottom equation by −3.

Next add the equations, and solve for y .

Substitute y = 10 into one of the original equations to find x .

You arrive at the same solution as before.

Graphing these two equations will help to illustrate what is happening.

Solving Application Problems Using the Elimination Method

Step 2: Subtract the second equation from the first.

Step 3: Solve this new equation for y .

Solution: x = 1, y = 2 or (1,2).

Now study some more worked examples:

Click on the buttons below to see how to solve these equations.

Substitute y into Equation 1 and solve for x: x + ( ) =

## IMAGES

## VIDEO

## COMMENTS

With this problem, there is no solution. If you multiply 3x + 2y = 18 by -2 (I chose -2 so when you add the equations together, variables cancel out), you

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Multiply for Elimination in Systems of Linear Equations. System of equations is {-7x. Image source: By Caroline Kulczycky. Report. Share. 1. Like.

The equations do not have any x or y terms with the same coefficient. ... In order to use the elimination method, you have to create variables that have the same

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We use the method of elimination to 'eliminate' one of the variables when solving a linear system, allowing us to solve for the other variable and in turn

Equations can be multiplied by a constant to allow for elimination by addition or subtraction. This process is called Elimination by Multiplication.

The elimination method for solving systems of linear equations uses the addition property of equality. You can add the same value to each side of an

The Elimination Method ... This method for solving a pair of simultaneous linear equations reduces one equation to one that has only a single variable. Once this