Introduction to the Substitution Method for Linear Equations

Algebra 1 Tutorial

Introduction to the Substitution Method for Linear Equations

Intro

The substitution method is one algebraic way to find the point where two lines intersect, or in other words, solve the system of equations. The first step to this is to solve for one of the variables, in this case we’ll call it “y” in one of the two equations. Next, you would SUBSTITUTE the equation you solved for “y” for the “y” in the equation you have, so far, left alone. Then, you will find that you only have one variable in the newly substituted equation and you are free to solve for it. Finally, once you have the solution for one variable, let’s call it x, you can plug that value into either one of the original equations to solve for y.
For example:
Solve for this system of equations:
2x+y=15 and y-x=3
First you would need to add x to both sides of the second equation in order to solve
for y. Now you are left with:
2x+y=15 and y=x+3
Now that you have solved for y in the second equation you can replace the y in the
first equation with”x+3″. Now you are left with:
2x+(x+3)=15
When you combine like terms you get:
3x+3=15
Now you subtract 3 from both sides:
3x=12
Isolate x by dividing 3 from both sides:
x=4
Now you have solved for x which means you have found the x-coordinate of the point
of intersection on the graph, so now you can easily find the y-coordinate, or solve
for y.
First, plug “x=4” into one of original equations. In this case, we will choose
“y-x=3”. Now you have:
y-4=3
Then, add 4 to both sides in order to solve for y. Now you have the solution:
y=7
By using substitution, you have solved for both x and y, finding that x=4 and y=7
which means that the point of intersection for the 2 linear equations is (4,7).

Sample Problem

Find the point of intersection for these two linear equations.
7y-3x=14 and 5x+4y=8

(8,3)


(2,0)


(5,7)


(0,2)


(-3,4)


Solution

First, solve for a variable in one of the equations. In this example, we’ll choose the first one. You should end up with:
y=(3/7)x+2
by adding 3x to both sides and then divide both sides by 7.

Next, plug that solution for “y” into the second equation which should give you:
5x+4((3/7)x+2)=8
Using the distributive property you can multiply both of the terms inside the parenthesis by 4 giving you:
5x+(12/7)x+8=8
By multiplying 5x by (7/7) you get (35/7)x which means you can now combine like terms giving you:
(47/7)x+8=8

Now, subtract eight from both sides giving you:
(47/7)x=0
Then, multiply both sides by 7 giving you
47x=0
Finally, solve for x by dividing both sides by 47 giving you:
x=0

Now you can plug in x into either equation. In this case, we’ll use the first one giving you:
7y-3(0)=14
Since 3(0)=0 you can simplify the equation to:
7y=14
By dividing both sides by 7 you get:
y=2
Now that you have the solutions for both the x and y coordinate you can form an ordered pair which is (0,2)



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