Download Solving Systems of Linear Equations by Substitution

7.2
Solving Systems of Linear Equations
by Substitution
Solving System of Linear Equations
by Substitution
• Solve the system.
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:
• I necessary, solve on of the equations for x or
y.
• Substitute the resulting expression for the
variable obtained in step 1 into the other
equation.
• Solve the equation, which is in now in one
variable.
• 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 by substitution:
3x = 4(6 – y)
4y + 3x = 24
• Since 3x = 4(6 – y) from the 1st equation,
this can be substituted in the 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.