Solving Linear Systems by Multiplying

Instructor: David Karsner

David holds a Master of Arts in Education

There are several ways to solve systems of linear equations: listing, substitution, graphing, and multiplying. This lesson will focus on how to solve them using multiplying (also known as elimination).

Jon and Jenn go on Vacation

The Family Debate
ColoradoFlorida

Jon and Jenn are siblings, and they get to choose the family vacation this year. Jon wants to go to the mountains, so he's thinking Colorado. Jenn wants to go to the beach, so she's thinking Florida. To keep family harmony, they choose to go to North Carolina, where they can visit the beach and the mountains. This scenario represents the logic that is behind solving a linear system. You have all the points that make one equation true, and all the points that make another equation true and you find the point that works for both of them. You can solve these systems by multiplying the equations. This lesson will explain how to use this process and give you a couple of examples to follow.

Systems of Linear Equations

A linear equation is a relationship between two variables with a degree of one.

Examples of linear equations:

y=4x-6

2x+3y=10

7y-105=x

A solution to a linear equation is a pairing of an x and y value that makes the equation true. The solution is given as a coordinate pair, such as (3,4). The equation y=4x-6 has an infinite number of solutions.

A few of the solutions to y=4x-6 are (4,10),(0-6), and (-2,-14); (10)=4(4)-6, (-6)=4(0)-6, and (-14)=4(-2)-6.

A system of linear equations are two or more linear equations that have been grouped to be solved together.

These two equations could be considered a system:

y=5x-8

2x+5y=30

A solution to a system of linear equations is a coordinate pair (x,y) that makes all of the equations in the system true.

For the system:

y=3x+1 and 2x+5y=22

The point (1,4) is a solution to the system.

(4)=3(1)+1 and 2(1)+5(4)=22

The standard form of an equation is when the equation looks like Ax+By=C. An example would be 3x+4y=20.

Multiplying Equations And Adding to Zero

One 'trick' or property of algebra that is most helpful here is that you can multiply all the terms of any equation by any number, and it will not change the solution(s) to that equation. For the equation, x+y=10, the points (1,9),(5,5), and (-2,12) are all solutions. Let's multiply the entire equation by 5, 5x+5y=50. The points (1,9),(5,5),(-2,12) are all still solutions. 5(1)+5(9)=50, 5(5)+5(5)=50, and 5(-2)+5(12)=50. Another property of algebra that is helpful when solving systems using multiplication is that opposite numbers and terms will add to zero. 7+(-7)= 0 and -2x+2x=0.

Solving Systems Using Multiplication

The key to using multiplying to solve linear systems is to find a number to multiply to one or both of the equations so that the x or y terms in one of the equations will have opposite coefficients from the x or y in the other equation. You would then add these two equations together. Since you now have opposite coefficients on one of the terms, that term will add to zero (fall away), leaving the other term and a number. This process is also called elimination because we are eliminating one of the variables.

Picking the Right Numbers
PickMultiply

Steps to Using Multiplying to Solve Linear Systems:

  1. Put both equations into standard form (Ax+By=C).
  2. Multiply one or both of the equations by a number so that you will have opposite coefficients on the same variable.
  3. Add the two equations together. Add the x's, the y's, and the numbers.
  4. Solve for the variable.
  5. Take your answer from the last step and plug it into of the equations (pick the one that looks easiest).
  6. Your solution will be in the form of a coordinate point (x,y).

Examples

Example One

Find the solution for the system of equations:

4x+3y=5, y=3x-7

Step One: Put in standard form.

4x+3y=5 (already in standard form)

y=3x-7 in standard form would be 3x-y=7

4x+3y=5 and 3x-y=7

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