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Chapter 5
Solving Systems of Linear
Equations
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
*See teacher or other
student class for work
on these examples
a. Solve a system of Equations by
graphing
y = -x + 8
y = 4x - 7
a. Solve a system of Equations by
graphing
x + 2y = 5
2x + 4y = 2
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.
*See teacher or other
student class for work on
these examples

Ex: graph the system
of inequalities
y<-x+4

Ex: Graph the system
of inequalities
x-y<-1
y 2x+3
x-y>3
5.2 Substitution
Solve Using Substitution
y = 3x
x + 2y = -21

Solve using Substitution
x + 5y = -3
x + 5y = -3
3x – 2y = 8
-5y -5y
Solve one

equation for a
variable
x + 2(3x)= -21
x + 6x = -21
7x = -21
/7
/7
x = -3
y = 3x
y = 3(-3)
Substitute 3x for y
Solve for x
Plug in -3 for x
and solve for y
y = -9
Solution = (-3, -9)
x = -3 – 5y
3(-3 – 5y) – 2y = 8
-9 -15y – 2y = 8
-9 -17y = 8
+9
+9
-17y = 17
/-17
/-17
y = -1
x = -3 -5(-1)
x = -3 +5
x=2
Solution = (2, -1)

Infinite or No solutions
6x – 2y = -4
y = 3x + 2
6x – 2(3x +2) = -4
6x – 6x - 4 = -4
-4 = -4
Infinite solutions
When all variables cancel, if:
the statement is true = infinite
solutions
the statement is false = no
solutions

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?
Yankees = x
Total games
Times games
5.2y + y = 31
6.2y = 31
/6.2 /6.2
y=5
x = 5.2(5)
x = 26
Reds = y
x + y = 31
x = 5.2y
Yankees = 26
Reds = 5
5.3 Elimination Using Addition
and Subtraction

Elimination: Addition
3x – 5y = -16
2x + 5y = 31
3x – 5y = -16
+ 2x + 5y = 31
5x = 15
/5
/5
x=3
3(3) – 5y = -16
9 – 5y = -16
-9
-9
-5y = -25
/-5
/-5
y=5
Add to eliminate
because the y’s
are the same
number
opposite signs
Solution:
(3, 5)

Elimination: Subtraction
5s + 2t = 6
9s + 2t = 22
5s + 2t = 6
- 9s + 2t = 22
-4s = -16
/-4 /-4
s=4
5(4) + 2t = 6
20 + 2t = 6
-20
-20
2t = -14
/2 /2
t =-7
Subtract to
eliminate because
the t’s are the
same number
same sign
Solution:
(4, -7)

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
4x - y = 12
+
2x + y = 18
4x - y = 12
6x = 30
/6
/6
x=5
2 (5) + y = 18
10 + y = 18
-10
-10
y=8
Add because
the y’s are the
same number
opposite signs
Solution:
5 and 8
5.4 Elimination Using
Multiplication

Multiply One Equation
3x + 4y = 6
5x + 2y = -4
3x +4y = 6
-2[5x + 2y = -4]
Solution:
(-2, 3)
Multiply one equation to
make a variable have the
same number and opposite
sign
3x +4y = 6
+ -10x + -4y = 8

Multiply Two Equations
3x + 4y = -25
2x – 3y = 6
3[3x +4y = -25]
4[2x – 3y = 6]
Multiply both equations to
make a variable have the
same number and opposite
sign
9x +12y = -75
+ 8x – 12y = 24
-7x = 14
/-7 /-7
17x = -51
/17 /17
x = -2
x = -3
3(-2) + 4y = 6
-6 + 4y = 6
+6
+6
4y = 12
/4
/4
y=3
Solution:
(-3, -4)
3(-3) + 4y = -25
-9 + 4y = -25
+9
+9
4y = -16
/4
/4
y = -4
5.5 Applying Systems of
Equations
Method
Graphing
The Best Time to Use
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
Elimination:
Subtraction
Elimination:
Multiplication
If one variable has coefficients with the same
number and opposite signs
If one variable has coefficients with the same
number and same sign
If none of the coefficients are the same number