1. Solving Systems of Linear Equations
Chapter 5
Solving Systems of Linear Equations

2. 5.1 Graphing Systems of Equations
Systems of equations- two equations together
A solution of a system of equations is an ordered pair that satisfies both equations
Consistent- the graphs of the equations intersect (at least one solution)
If consistent with exactly 1 solution = independent
If consistent with infinite solutions = dependent
Inconsistent- the graphs of the equations are parallel
No ordered pair solutions
Exactly one solution
Infinite solutions
No solutions
Consistent and Independent
Consistent and Dependent
Inconsistent

3. *See teacher or other student class for work on these examples
a. Solve a system of Equations by graphing
a. Solve a system of Equations by graphing
y = -x + 8
y = 4x - 7
x + 2y = 5
2x + 4y = 2

4. 6.8 Graphing Systems of Inequalities
1. get the inequality in slope-intercept from
2. State the slope and y-intercept
3. Graph the intercept and use slope to find the next points
4. Draw the line:
< or > = dotted, or = solid
5. Test an ordered pair not on the line
-if true, shade that side of the line
-if false, shade the other side of the line
6. Repeat steps 1-5 for the second inequality.

5. *See teacher or other student class for work on these examples
Ex: graph the system of inequalities
y<-x+4
y 2x+3
Ex: Graph the system of inequalities
x-y<-1
x-y>3

6. 5.2 Substitution x + 2(3x)= -21 Substitute 3x for y
Solve Using Substitution
y = 3x
x + 2y = -21
Solve using Substitution
x + 5y = -3
3x – 2y = 8
x + 5y = -3
-5y -5y
x = -3 – 5y
Solve one equation for a variable
x + 2(3x)= -21
Substitute 3x for y
3(-3 – 5y) – 2y = 8
x + 6x = -21
-9 -15y – 2y = 8
7x = -21
Solve for x
-9 -17y = 8
/ /7
-17y = 17
x = -3
/ /-17
y = -1
y = 3x
Plug in -3 for x and solve for y
y = 3(-3)
x = -3 -5(-1)
y = -9
x = -3 +5
Solution = (2, -1)
x = 2
Solution = (-3, -9)

7. Infinite or No solutions Write and solve a System of Equations
6x – 2y = -4
y = 3x + 2
Write and solve a System of Equations
The New York Yankees and Cincinnati Reds together have won a total 31 World Series. The Yankees have won 5.2 times as many as the Reds. How many have each team won?
6x – 2(3x +2) = -4
Yankees = x Reds = y
6x – 6x - 4 = -4
-4 = -4
Total games x + y = 31
Infinite solutions
Times games x = 5.2y
5.2y + y = 31
When all variables cancel, if:
the statement is true = infinite solutions
the statement is false = no solutions
6.2y = 31
Yankees = 26
Reds = 5
/ /6.2
y = 5
x = 5.2(5)
x = 26

8. 5.3 Elimination Using Addition and Subtraction
Elimination: Addition
3x – 5y = -16
2x + 5y = 31
Elimination: Subtraction
5s + 2t = 6
9s + 2t = 22
Subtract to eliminate because the t’s are the same number same sign
Add to eliminate because the y’s are the same number opposite signs
3x – 5y = -16
+ 2x + 5y = 31
5s + 2t = 6
- 9s + 2t = 22
5x = 15
-4s = -16
/ /5
Solution:
(4, -7)
Solution:
(3, 5)
/ /-4
x = 3
s = 4
3(3) – 5y = -16
5(4) + 2t = 6
9 – 5y = -16
20 + 2t = 6
-5y = -25
2t = -14
/ /-5
/2 /2
y = 5
t =-7

9. Write and solve a system of equations
Twice one number added to another number is 18. Four times the first number minus the other number is 12. Find the numbers.
2x + y = 18
2x + y = 18
Add because the y’s are the same number opposite signs
4x - y = 12 +
4x - y = 12
6x = 30
/ /6
x = 5
Solution:
5 and 8
2 (5) + y = 18
10 + y = 18
y = 8

10. 5.4 Elimination Using Multiplication
Multiply One Equation
3x + 4y = 6
5x + 2y = -4
Multiply Two Equations
3x + 4y = -25
2x – 3y = 6
Multiply one equation to make a variable have the same number and opposite sign
Multiply both equations to make a variable have the same number and opposite sign
3x +4y = 6
-2[5x + 2y = -4]
3[3x +4y = -25]
4[2x – 3y = 6]
9x +12y = -75
8x – 12y = 24
3x +4y = 6
-10x + -4y = 8 + +
17x = -51
-7x = 14
/17 /17
/ /-7
x = -3
x = -2
Solution:
(-2, 3)
Solution:
(-3, -4)
3(-3) + 4y = -25
3(-2) + 4y = 6
-9 + 4y = -25
-6 + 4y = 6
4y = -16
4y = 12
/ /4
/ /4
y = -4
y = 3

11. 5.5 Applying Systems of Equations
Method
The Best Time to Use
Graphing
To estimate the solution. When both equations are in Slope-Intercept Form
Substitution
If one variable in either equation has a coefficient of 1
Elimination: Addition
If one variable has coefficients with the same number and opposite signs
Elimination:
Subtraction
If one variable has coefficients with the same number and same sign
Multiplication
If none of the coefficients are the same number