Solving Systems of Equations Using
Substitution
Lesson 6-2
The solution of a system of equation scan be found using one of 3 methods:
substitution
graphing
________________________
, _________________________,
or by
elimination
_________________________
(The purpose of the substitution
method is that once you find one of the values (either x or y), you can then
substitute it into either of the original equations to find the other value.)
Ex. A
x = 4y
4y
4y
4x – y = 75 (Since x = ______., ________ can be substituted
everywhere there is an “x” in the second equation.)
4x – y = 75
Second equation
4(4y) – y = 75
____________
Substitute “4y” for “x”
16y – y = 75
_____________
Simplify
15y = 75
_____________
Combine like terms
15y /15 = 75/15
_____________
y= 5
_____________
Divide both sides
Simplify
NOW use x = 4y to find the
value of x. (Substitute the
answer you just got for “4”)
x = 4y
x =____________
4(5)
___
Substitute
x = 20
_______________
Simplify
The solution is ___________
(Don’t forget to write the solution, which is an ordered
(5, 20)
pair.)
Ex. B Solve
y = 2x
3x + 4y = 11
3x + 4y = 11
________________
3x + 4(2x) = 11
________________
Second equation
3x + 8x = 11
_______________
11x = 11
_______________
Substitute “2x” for “y”
Use the first equation to find
the value of x.
Simplify
y = 2x
_______________
1st equation
Combine like terms
y = 2(1)
_______________
Substitute
y=2
_______________
Simplify
11x/11= 11/11
_________________________
Divide both sides
x=1
____________________
Simplify
(1, 2)
The solution is ___________
PRACTICE
1
x = 2y
4x + 2y = 15
_____________________
____________________
____________________
____________________
____________________
_____________________
________________
________________
________________
The solution is ___________
PRACTICE
2
y = 3x - 8
y=4–x
_____________________
____________________
____________________
____________________
____________________
_____________________
________________
________________
________________
The solution is ___________
PRACTICE:
3
2x + 7y = 3
x = 1 – 4y
_____________________
____________________
____________________
____________________
____________________
_____________________
________________
________________
________________
The solution is ___________
4x + y = 12
4x – 4x + y = 12 – 4x
y = 12 – 4x
-2x – 3y = 14
-2x – 3(12 – 4x) = 14
-2x – 36 + 12x = 14
10x – 36 = 14
10x – 36+36 = 14 + 36
4x + y = 12
4(5) + y = 12
20+ y = 12
20+ y – 20 = 12 – 20
y = -8
10x = 50
10x/10 = 50/10
x=5
(5, -8)
You can check your answer using the graphing
calculator OR by substituting the ordered pair
into each of the equations to see if it is a
solution for each equation.
2x + 2y = 8
2(2 – y) + 2y = 8
x+y=2
x+y-y=2-y
4 – 2y + 2y = 8
4=8
x=2–y
4=8
false
no solutions
parallel lines
-6x +4y = -6
3x – 2y = 3
3x – 2y + 2y = 3 + 2y
3x = 3 + 2y
(3x)/3 = (3 + 2y)/3
x = (3 + 2y)/3
-2(3+ 2y) +4y = -6
-6 – 4y +4y = -6
-6 = -6
-6 = -6
The statement ___________
is _________.
There are
true
infinitely
many
______
_______________________.
The graphs of the lines
the same.
are ________________________
The school bookstore sells T-shirts for $8 and sweatshirts for $12. Last month,
the store sold 37 T-shirts and sweatshirts for a total of $376. How many T-shirts
were sold?
x = number of T-Shirts sold
Y = number of sweatshirts sold
x + y = 37
8x + 12y = 376
Equation based on the
number of items sold
Equation based on the
cost of items sold
x + y = 37
y = 37 - x
8x + 12 y = 376
8x + 12 (37 – x) = 376
8x + 444 – 12x = 376
-4x + 444 = 376
-4x + 444 – 444 = 376 – 444
-4x = -68
x = 17
17 T-shirts were sold.
(so 20 sweatshirts were sold)
Check:
17(8 ) + 12(20) = 136 + 240 = 376
Watch the following videos for additional information on
solving systems of equations using substitution.
Solving systems using substitution
Using Systems of Equations to Solve Real World Problems