1. The student will be able to:
Objective
The student will be able to:
Solve Systems of Equations using substitution.
Adapted from Skip Tyler PowerPoint

2. Solving Systems of Equations
You can solve a system of equations using different methods. The idea is to determine which method is easiest for that particular problem.
These notes show how to solve the system algebraically using SUBSTITUTION.
You will usually use this method mostly when one of your equations is ALREADY solved for a variable, or easily solved for a variable with one transformation.

3. Solving a system of equations by substitution
Step 1: Solve an equation for one variable.
Pick the easier equation. The goal
is to get y= ; x= ; a= ; etc.
Step 2: Substitute
Substitute the equation solved in Step 1 into the other equation.
Step 3: Solve the equation.
Get the variable by itself.
Step 4: Plug back in to find the other variable.
Substitute your answer back into
the equation to find other value.
Step 5: Check your solution.
Substitute ordered pair
(solution you just found)
into BOTH equations.

4. 1) Solve the system using substitution(EASY)
x + y = 5
y = 3 + x
Step 1: Solve an equation for one variable.
The second equation is
already solved for y!
Step 2: Substitute
x + y = 5 becomes
x + (3 + x) = 5
2x + 3 = 5
2x = 2
x = 1
Step 3: Solve the equation.

5. 1) Solve the system using substitution(EASY)
x + y = 5
y = 3 + x
Remember, we just found that x = 1
x + y = 5
(1) + y = 5
y = 4
Step 4: Substitute back in to find the other variable.
(1, 4)
(1) + (4) = 5
(4) = 3 + (1)
Step 5: Check your solution.
The solution is (1, 4). What do you think the answer would be if you graphed the two equations?

6. 1) Solve the system using substitution(EASY)
Try it before you click for answer. You ONLY get better by practicing problems.
HINT: Put 6 + y in for x
1) Solve the system using substitution(EASY)
3x + 3y = 6
x = 6 + y
Step 1: Solve an equation for one variable.
The second equation is
already solved for x!
Step 2: Substitute
3x + 3y = 6 becomes 3(6+y)+ 3y = 6
18 + 3y + 3y = 6
18 + 6y = 6
6y = -12
y = -2
Step 3: Solve the equation.

7. 1) Solve the system using substitution(EASY)
3x + 3y = 6
x = 6+ y
We found y= -2 in previous step.
x= 6+y
x= 6 + (-2)
x = 4 , solution ( 4, -2)
Step 4: Plug back into easiest equation to find the other variable.
(4, -2)
3(4) + 3(-2) = 6
(4) = 6 + (-2)
Step 5: Check your solution into each equation.
The solution is (4, -2). What do you think the answer would be if you graphed the two equations?

8. Which answer checks correctly?
Try it before you click for answer. You ONLY get better by practicing problems.
HINT: Put 4y-17 in for x
Which answer checks correctly?
3(4y - 17) –y = 4
12y y = 4
11y = 55 , y = 5
Substitute 5 in for y in one of the equations to find x.
x = 4(5) -17 >>> x = =3
Solution: (3, 5)
3x – y = 4
x = 4y - 17
(2, 2)
(5, 3)
(3, 5)
(3, -5)

9. Which answer checks correctly?
Try it before you click for answer. You ONLY get better by practicing problems.
HINT: Put 4x-4 in for y
Which answer checks correctly?
2x + 2(4x - 4) = 12
2x + 8x -8 = 12
10x = 20 , x = 2
Substitute 2 in for x in one of the equations to find y.
y = 4(2) -4 >>> y = = 4
Solution: (2, 4)
2x + 2y = 12
y = 4x - 4
(2, 4)
(4, 2)
(-2, -4)
(-4, -2)

10. 2) Solve the system using substitution
3y + x = 7
4x – 2y = 0
Step 1: Solve
an equation for one
Variable (get x or y by itself). Choose the equation that will transform
the easiest.
It is easiest to solve the
first equation for x.
3y + x = 7
-3y y
x = -3y + 7
Step 2: Substitute
4x – 2y = 0 becomes
4(-3y + 7) – 2y = 0

11. 2) Solve the system using substitution
Remember, we were here
4(-3y + 7) – 2y = 0
3y + x = 7
4x – 2y = 0
-12y + 28 – 2y = 0
-14y + 28 = 0
-14y = -28
y = 2
Step 3: Solve the equation.
4x – 2y = 0
4x – 2(2) = 0
4x – 4 = 0
4x = 4
x = 1
Step 4: Substitute back in to find the other variable. (Either equation, doesn’t matter)

12. 2) Solve the system using substitution
3y + x = 7
4x – 2y = 0
Step 5: Check your solution.
(1, 2)
3(2) + (1) = 7
4(1) – 2(2) = 0
When is solving systems by substitution easier to do than graphing?
When only one of the equations has a variable already isolated (like in example #1).

13. 2x + 4y = 4 3x + 2y = 22 -4y + 4 -2y + 2 -2x + 4 -2y+ 22
If you solved the first equation for x, what would be substituted into the bottom equation.
2x + 4y = 4
-4y = -4y
2x = -4y + 4
x = -2y + 2
2x + 4y = 4
3x + 2y = 22
-4y + 4
-2y + 2
-2x + 4
-2y+ 22

14. Solve the system using substitution. Special Case #1
x = 3 – y
x + y = 7
Step 1: Solve an equation for one variable.
The first equation is
already solved for x!
Step 2: Substitute
x + y = 7
(3 – y) + y = 7
3 = 7
The variables were eliminated!!
This is a special case.
Does 3 = 7? FALSE!
Step 3: Solve the equation.
When the result is FALSE, the answer is NO SOLUTIONS.

15. Solve the system using substitution Special Case #2
2x + y = 4
4x + 2y = 8
Step 1: Solve an equation for one variable.
The first equation is
easiest to solved for y!
y = -2x + 4
4x + 2y = 8
4x + 2(-2x + 4) = 8
Step 2: Substitute
4x – 4x + 8 = 8
8 = 8
This is also a special case.
Does 8 = 8? TRUE!
Step 3: Solve the equation.
When the result is TRUE, the answer is INFINITELY MANY SOLUTIONS.