1. Solving Systems of Equations By Substitution – Easier
Dr. Fowler  CCM
Solving Systems of Equations By Substitution – Easier

2. 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
Put 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 the value of the variable
into the equation.
Step 5: Check your solution.
Substitute your ordered pair into
BOTH equations.

3. EXAMPLE 1 Solve by substitution:
The second is solved for X. Substitute this into OTHER equation for X:
Substitute found y into other equation:
The solution set found by the substitution method will be the same as the solution found by graphing. The solution set is the same; only the method is different. ALWAYS put answer in Alphabetical order. (x,y)

4. 2) Solve the system using substitution
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 x + (3 + x) = 5
2x + 3 = 5
2x = 2
x = 1
Step 3: Solve the equation.

5. 2) Solve the system using substitution
x + y = 5
y = 3 + x
x + y = 5
(1) + y = 5
y = 4
Step 4: Plug 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. 3) Solve the system using substitution
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.

7. 4) Solve the system using substitution
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.

8. Example 5) Solve the following system of equations using the substitution method.
y = 3x – 4 and 6x – 2y = 4
The first equation is already solved for y. Substitute this into second equation.
6x – 2y = 4
6x – 2(3x – 4) = 4 (substitute)
6x – 6x + 8 = 4 (use distributive property)
8 = (simplify the left side) Does 8=4? FALSE.
Examples like this – the answer is NO SOLUTION Ø. If you graphed them, they would be PARALLEL LINES.

9. EXAMPLE 6 Solve the system by the substitution method.
The second is solved for X. Substitute this into OTHER equation for X:
Substitute found y into other equation:

10. Example #7:
y = 4x
3x + y = -21
Step 1: Solve one equation for one variable.
y = 4x (This equation is already solved for y.)
Step 2: Substitute the expression from step one into the other equation.
3x + y = -21
3x + 4x = -21
Step 3: Simplify and solve the equation.
7x = -21
x = -3

11. y = 4x
3x + y = -21
Step 4: We found x = -3. Now, substitute this into either original equation to find y:
y = 4x (easiest)
y = 4(-3)
y = -12
Solution to the system is (-3, -12).

12. Excellent Job !!! Well Done

13. Stop Notes
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