1. Section 3.2 - Solving Linear Systems Algebraically
ALGEBRA TWO
CHAPTER THREE: SYSTEMS OF LINEAR EQUATIONS AND INEQUALITIES
Section Solving Linear Systems Algebraically

2. LEARNING GOALS
Goal One - Use algebraic methods to solve linear systems.
Goal Two - Use linear systems to model real-life situations.

3. THE SUBSTITUTION METHOD
ALGEBRAIC SOLUTIONS
THE SUBSTITUTION METHOD
STEP 1: Solve one of the equations for one of its variables.
STEP 2: Substitute the expression from Step 1 into the other equation and solve for the other variable.
STEP 3: Substitute the value from Step 2 into the revised equation from Step 1 and solve. 1

4. STEP 1: Solve one of the equations for one variable.
SUBSTITUTION METHOD
Use the substitution method to solve the linear system:
Equation x + y = -2
Equation x - 3y = 17
STEP 1: Solve one of the equations for one variable.
6x + y = -2
Write original equation
y = -6x -2
Subtract 6x from each side. 1

5. STEP 2: Substitute into other equation.
SUBSTITUTION METHOD
Use the substitution method to solve the linear system:
Equation x + y = -2
Equation x - 3y = 17
STEP 2: Substitute into other equation.
6x + y = -2
Write original equation
y = -6x -2
Subtract 6x from each side.
4x - 3(-6x - 2) = 17
Substitute into other equation.
4x + 18x + 6 = 17
Use distributive property
22x = 11
Combine like terms.
x = 1/2
Simplify 1

6. SUBSTITUTION METHOD STEP 3: Substitute into revised equation (Eq 1).
Use the substitution method to solve the linear system:
Equation x + y = -2
Equation x - 3y = 17
STEP 3: Substitute into revised equation (Eq 1).
y = -6x - 2
Write revised equation
y = -6(1/2) - 2
Substitute 1/2 for x.
y = = -5
Simplify
y = -5
The solution to this linear system is (1/2 , -5). 1

7. STEP 1: Solve one of the equations for one variable.
SUBSTITUTION METHOD
Use the substitution method to solve the linear system:
Equation x - y = 6
Equation x + 2y = -9
STEP 1: Solve one of the equations for one variable. 1

8. STEP 1: Solve one of the equations for one variable.
SUBSTITUTION METHOD
Use the substitution method to solve the linear system:
Equation x - y = 6
Equation x + 2y = -9
STEP 1: Solve one of the equations for one variable.
2x - y = 6
Write original equation
-y = -2x + 6
Subtract 2x from each side.
y = 2x - 6
Multiply by -1. 1

9. STEP 2: Substitute into other equation.
SUBSTITUTION METHOD
Use the substitution method to solve the linear system:
Equation x - y = 6
Equation x + 2y = -9
STEP 2: Substitute into other equation.
2x + 2(2x - 6) = -9
Substitute into other equation.
2x + 4x - 12 = -9
Use distributive property
6x = 3
Combine like terms.
x = 1/2
Simplify 1

10. STEP 3: Substitute into revised equation.
SUBSTITUTION METHOD
Use the substitution method to solve the linear system:
Equation x - y = 6
Equation x + 2y = -9
STEP 3: Substitute into revised equation.
y = 2x - 6
Write revised equation
y = 2(1/2) - 6
Substitute 1/2 for x.
y = = -5
Simplify
y = -5
The solution to this linear system is (1/2 , -5). 1

11. THE LINEAR COMBINATION METHOD
ALGEBRAIC SOLUTIONS
THE LINEAR COMBINATION METHOD
STEP 1: Multiply one or both of the equations by a constant to obtain coefficients that differ only in sign for one of the variables.
STEP 2: Add the revised equations from Step 1. Combining like terms will eliminate one of the variables. Solve for the remaining variable.
STEP 3: Substitute the value from Step 2 into either one of the original equations and solve for the other variable. 1

12. Use the linear combination method to solve the linear system:
Equation x + 3y = 3
Equation x + 4y = 4
STEP 1: Multiply by a constant.
6x + 3y = 3
times 4
24x + 12y = 12
8x + 4y = 4
times -3
-24x - 12y = -12
STEP 2: Add the equations.
0 = 0
Because the statement 0 = 0 is always true, there are infinitely many solutions. 1

13. Use the linear combination method to solve the linear system:
Equation x + 2y = 0
Equation x - 5y = 17
STEP 1: Multiply by a constant.
9x + 2y = 0
times 1
9x + 2y = 0
3x - 5y = 17
times -3
-9x + 15y = -51
17y = -51
STEP 2: Add the equations.
y = -51/17
y = -3 1

14. Use the linear combination method to solve the linear system:
Equation x + 2y = 0
Equation x - 5y = 17
STEP 3: Substitute into either original equation.
9x + 2y = 0
Write equation
The solution to the linear system is (2/3 , -3)
9x + 2(-3) = 0
Substitute -3 for y.
9x + (-6) = 0
Multiply
9x + (-6) + 6 = 0 + 6
Add 6 to each side
9x/ 9 = 6/9
Divide each side by 9
x = 2/3
Simplify 1

15. Use the linear combination method to solve the linear system:
Equation x - 3y = 5
Equation x + 9y = 12
STEP 1: Multiply by a constant.
2x - 3y = 5
times 3
6x - 9y = 15
-6x + 9y = 12
times 1
-6x + 9y = 12
0 = 27
STEP 2: Add the equations.
Because the statement 0 = 27 is never true, there are no solutions to this system. 1

16. LINEAR COMBINATION METHOD STEP 1: Multiply by a constant.
Use the linear combination method to solve the linear system:
Equation x + 6y = 1
Equation x + 2y = -3
STEP 1: Multiply by a constant.
11x + 6y = 1
3x + 2y = -3 1

17. Use the linear combination method to solve the linear system:
Equation x + 6y = 1
Equation x + 2y = -3
STEP 1: Multiply by a constant.
11x + 6y = 1
times 1
11x + 6y = 1
3x + 2y = -3
times -3
-9x - 6y = 9
2x = 10
STEP 2: Add the equations.
x = 5 1

18. Use the linear combination method to solve the linear system:
Equation x + 6y = 1
Equation x + 2y = -3
STEP 3: Substitute into either original equation.
3x + 2y = -3
Write equation
The solution to the linear system is (5, -9)
3(5) + 2y = -3
Substitute 5 for x.
15 + 2y = -3
Multiply
15 + 2y - 15 =
Subtract 15
2y/ 2 = -18/2
Divide each side by 2
y = -9
Simplify 1

19. ASSIGNMENT
READ & STUDY: pg
WRITE: pg #11, #15, #21, #23, #27, #33, #35, #41, #49