To Be Equal Means That the Two Expressions Are the Same Or Have the Same Value

Algebra I
Solving Systems of Equations Using Substitution

NAME: _____KEY______

Today we will be learning how to solve a system of equations using a method known as substitution. Rather than simply listening to me explain how to do it, I’m going to have you and a partner explore and discover the method yourselves. Woo-hoo! 

Let’s begin …

EXAMPLE 1

y = x + 1

2x + y = 7

Explain:

In the system of equations above, the first equation is “y = x + 1.”
Explain what it means for two expressions to be equal.

To be equal means that the two expressions are the same or have the same value.

Observe and discover:

y = x + 1
2x + x + 1 = 7
2x + y= 7

The system of equations is written on the left.
The first step of the substitution method is on the right.

Explain WHAT was done and WHY it would be OK to do that.

The variable y in the 2nd equation was replaced by the expression that was equal to y in the first equation.

Begin to solve:

Using the equation from step 1 of the substitution method (also written below), solve for the variable. (Show your work!)

2x + x + 1 = 7

x = ____2_____

Finish solving:

y = x + 1

2x + y = 7

Select either equation from the system and write it here: ___y = x + 1______
Using your answer from the “Begin to Solve” step above, how could you find the value of y?
Explain in words.

You could substitute into the equation the number you found for x and then solve for y.

Now actually do the work you just described and find the value of y. (Remember to show your work!)

y = 2 + 1

y = 3

Let’s see if it would make a difference if we had chosen the other equation from the system to find y.
Write down the equation you didn’t pick … follow the same process … and find the value of y again.

2x + y = 7

2(2) + y = 7

4 + y = 7

y = 3

Check:

State the values you found for x and y as an ordered pair (x, y):____(2, 3)______

Substitute those values for x and y into both equations and simplify.

y = x + 12x + y = 7

3 = 2 + 12(2) + 3 = 7

3 = 34 + 3 = 7

7 = 7

If the result for both equations is a true statement (examples: 5 = 5 … 24= 24 … -2 = -2 … etc.), that means our ordered pair IS a solution to the system because it satisfies (makes true) both equations. Sweet. 

Confirm:

Let’s graph the system of equations to confirm our solution from a graphical perspective.

y = x + 1

slope = __1__

y-intercept = __1__

2x + y = 7
Solve for y to get in slope-intercept form

y = -2x + 7

slope = __-2__

y-intercept = __7__

Graph the 2 equations on the grid above.
State the intersection point: ___(2, 3)____

How does this compare to the solution you got using substitution?They are the same! 

Substitution – Show Me What Ya’ Got NAME: ______
(Individual Work)

Explain each step in solving this system of equations using substitution.

y = 4x + 2ORIGINAL SYSTEM OF EQUATIONS
3y – 2x = 26

Explain in words what’s been done in each step.

3(4x + 2) – 2x = 26The value of y in the 2nd equation has been replaced by the expression equal to y in the 1st equation.

12x + 6 – 2x = 26The distributive property was used here.
10x + 6 = 26
- 6 - 6
10x = 20The equation was solved for x.
10 10
x = 2

y = 4x + 2 OR3y – 2x = 26The value found for x was substituted into
y = 4(2) + 2 3y – 2(2) = 26each of the equations to find the value of y.
y = 10 3y – 4 = 26
+ 4 + 4
3y = 30
3 3
y = 10

SOLUTION:(2, 10)

Solve these systems of equations using substitution.

(1)y = 4x(2)3x – 2y = 5
3y + 2x = 28x = 15 – y

(2, 8)(7, 8)

Substitution – Show Me What Ya’ Got (continued)
(Back to partners)

Lastly … let’s shake things up a bit and turn them on their heads. Fun! 

(I)SHAKE-UP # 1

The following system of equations is not quite ready to be solved using substitution.
How is this system different compared to the other systems we’ve looked at so far where we HAVE used the substitution method?

y – x = - 2Neither equation is solved for one of the variables.
x + y = 12(Neither equation has one variable by itself.)

How could you “fix” one of the equations so that we COULD use substitution easily? Answer in words, and then do so (actually “fix” it). You do not need to solve the system.

Solve one of the equations for x or for y.

(II)SHAKE-UP # 2

ORIGINAL SYSTEM

2x + 5y = 16 2x + 5y = 16Finish the problem for me:
2x + y = 4 2x + 5(4 – 2x) = 162x + y = 4
2x + 20 – 10x = 162( 1/2 ) + y = 4

2x + y = 4 -8x + 20 = 16y = 3
-2x - 2x -20 -20
y = 4 – 2x -8x = -4
-8 -8SOLUTION: __(1/2, 3)___
x = __1/2___
(How is this value of x different from the others we’ve found? Is that OK?)

It’s a fraction. Is that OK? OF COURSE IT IS! 

(III)SHAKE-UP # 3

ORIGINAL SYSTEM

y = 4x – 5 8x -2y = 20
8x – 2y = 20 8x - 2(4x – 5) = 20
8x – 8x + 10 = 20
10 = 20

Say What?!

If I solved the equation correctly, and I ended up with a FALSE statement, what must that imply about the solution? If you are unsure, use your graphing calculator to graph the equations. See if that gives you the clue you need. 

A false statement implies that there is no solution to the system.

(IV)SHAKE UP # 4

ORIGINAL SYSTEM

y = 2x – 1 3y + 3 = 6x
3y + 3 = 6x 3(2x – 1) + 3 = 6x
6x – 3 + 3 = 6x
6x = 6x
- 6x- 6x
0 = 0

That’s true, but what the heck?!

If I solved the equation correctly, and I ended up with a TRUE statement, what must that imply about the solution? If you are unsure, use your graphing calculator to graph the equations. See if that gives you the clue you need. 

A true statement implies that there are infinitely many solutions.