Systems of Linear Equations: Substitution

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Systems of Linear Equations: Substitution

Introduction:

A system of equations simply means that you are comparing multiple equations. In this lesson we will look at how to solve (find the intersection of) two linear equations using substitution.

The method of substitution is based on the principle of equivalency. What this means is that the definition of any variable in a system will be consistent throughout the system. In other words the

$x$ in one equation will be equivalent to

$x$ in any other equation which is part of the same group of equations.

If a system of equations consists of

$y=3x+5$ and

$y=x+15$ then

$3x+5=x+15$ because the

$y$ in both equations are equal therefore the equations are equivalent.

If we subtract

$x$ from both sides and then subtract

$5$ from both sides we get

$2x=10$ .

Dividing by

$2$ results in

$x=5$ .

We can test this by replacing

$x$ with

$5$ . This results in

$20=3(5)+5$ and

$20=(5)+15$ which means that

$x=5$ and

$y=20$ is true for both equations.

This gives a solution of

$(5,20)$ which is the point at which both lines intersect.

Another example would be the system of

$y=3x-12$ and

$x=y-2$ . In this case we will substitute the

$x$ in one equation for the definition of

$x$ in the other equation. This removes the

$x$ variable and we can solve for

$y$ .

$y=3(y-2)-12$

$y=3y-6-12$

$-2y=-18$

$y=9$

We canthen put that value of y back into either equation and find the value of

$x$ .

$x=(9)-2$

$x=7$

This means that the solution for this system of equations is

$(7,9)$

Directions for This Lesson:

Watch the video below, do the practice questions, and then complete the worksheet.

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Additional Resources:

Linear Systems by Substitution

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Systems of Linear Equations: Substitution

Systems of Linear Equations: Substitution