Download 3.2 Solve Linear Systems Algebraically

3.2 Solve Linear Systems
Algebraically
Algebra II
Substitution Method
1) Solve one of the equations for one of its
variables.
2) Substitute the expression from Step 1
into the other equation and solve for the
other variable.
3) Substitute the value from Step 2 into the
revised equation from Step 2 and solve.
Example – Substitution Method
2x + 5y = -5  equation 1
x + 3y = 3
 equation 2
Step 1) Solve equation 2 for x
x = -3y + 3
Step 2) Substitute into equation 1 and solve for y
2 (-3y + 3) + 5y = -5
y = 11
Step 3) Substitute the value of y into revised equation 2 and
solve for x.
x = -3 (11) +3
x = -30
Solution: (-30, 11)
CHECK!
Solve this system:
Elimination Method
1) Multiply one or both of the equations by
a constant to obtain coefficients that
differ only in sign for one of the variables.
2) Add the revised equations from Step 1.
Combining like terms will eliminate one
of the variables. Solve for the remaining
variable.
3) Substitute the value obtained in Step 2
into either of the original equations and
solve for the other variable.
Example - Elimination Method
8x + 2y = 4
 equation 1
-2x + 3y = 13  equation 2
Multiply equation 2 by 4 so coefficients of x differ only
in sign.
-8x + 12 y = 52
Add the revised equations for x and solve.
8x + 2y = 4
- 8x + 12 y = 52
14y = 56
y=4
Substitute the value of y into one of the original
equations and solve for x.
8x + 2(4) = 4
x = -.5
Solution: (-.5, 4)
Solve this system:
Step 1)
Step 2)
Step 3)
Example 3
Solve this linear system:
x – 2y = 4
3x – 6y = 8
 equation 1
 equation 2
Because the coefficient of x in the first equation is, use the
substitution method.
Step 1) Solve for x:
Step 2) Substitute:
x = 2y + 4
3(2y + 4) – 6y = 8
6y + 12 – 6y = 8
12 = 8
Because the statement is never true, there is no solution
Example 4
Solve this linear system:
Step 1)
4x – 10y = 8
 equation 1
-14x + 35y = -28  equation 2
4x – 10y = 8
 x7 
-14x + 35y = -28  x2 
Step 2) Add
28x – 70y = 56
-28x + 70y = -56
0=0
Because the statement 0 = 0 is always true, there are
infinitely many solutions
Homework
• Page 164-167/ 4-14 even, 56, 72