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7.2
Solving Systems of Linear Equations
by Substitution
Solving System of Linear Equations
by Graphing
• Solve the system of equation by graphing.
a) y = 3x – 2
b) 2x + y = 8
• a) y = 3x – 2
b) y = -2x + 8
line b)
line a)
(0, 8)
(1, 6)
(2, 4)
(1, 1)
(0, -2)
Solving a system of linear equations
by substitution
•Solve the system
y = 3x – 2
2x + y = 8
•We note that from the first equation y = 3x – 2.
Substitute this for y in the second equation.
2x + y = 8
2x + (3x – 2) = 8
2x + 3x – 2 = 8
5x = 10
x=2
y = 3x – 2 = 3(2) – 2 = 4
Thus, x = 2 and y = for, or (2, 4)
Solving a system of linear equations
by substitution
To solve a system of equations in x and y by the
substitution method:
1. If necessary, solve one of the equations for x
or y.
2. Substitute the resulting expression for the
variable obtained in step 1 into the other
equation.
3. Solve the equation, which is in now in one
variable.
4. Solve for the other variable using this result.
Your Turn
• Solve the system by substitution:
2x + y = -5
3x + 5y = -4
(Hint: Use the first equation to obtain an expression for y.)
• Solution:
2x + y = -5 → y = -2x – 5
3x + 5y = -4
3x + 5(-2x – 5) = -4
3x – 10x - 25 = -4
-7x = 21 → x = -3
y = -2x – 5
y = -2(-3) – 5 → y = 1
Thus, x = -3, y = 1; or (-3, 1)
Your Turn
• Solve the system by substitution.
x = 4(3 – y)
2x = 4(3 – 2y)
• Since x = 4(3 – y), substitute this for x in the second
equation.
2x = 4(3 – 2y)
2(4(3 – y)) = 4(3 – 2y)
2(12 – 4y) = 4(3 – 2y)
24 – 8y = 12 – 8y
24 = 12
• This false, for any combination of (x, y). No solution!
Or, solution set is ∅.
Your Turn
• Solve the system of equations by using the
substitution method.
3x + y = 0
5x + 2y = -1
• Solution
From 1st equation, y = -3x
Substituting this in 2nd equation,
5x + 2(-3x) = -1
-x = -1
x=1
Since y = -3x, y = -3(1) = -3
Solution: (1, -3)
Your Turn
Solve the system by substitution:
a) 3x = 4(6 – y)
b) 4y + 3x = 24
Hint: Note that 3x in b) can be substituted by 3x in a).
• From 1st equation, 3x = 4(6 – 7). Substitute this in 2nd
equation.
4y + 3x = 24
4y + 4(6 – y) = 24
4y + 24 – 4y = 24
24 = 24
• This is true for any (x, y) combination.
Thus, there are infinite sets of (x, y) combinations for this
system. The equations represent the same line.
Your Turn
Solve the system of equations using the substitution
method.
3x – y = 7
2x + 3y = 1
Solution:
From 1st equation, 3x – 7 = y
From 2nd equation, 2x + 3(3x – 7) = 1
2x + 9x – 21 = 1
11x = 22, and x = 2
Since y = 3x – 7, y = 3(2) – 7 = -1
Solution: (2, -1)