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SOLVING SYSTEMS OF
EQUATIONS AND
INEQUALITIES
Module 1 Lesson 5
What is a system of equations?
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

A system of equations is when you have two or
more equations using the same variables.
The solution to the system is the point that satisfies
ALL of the equations. This point will be an ordered
pair.
When graphing, you will encounter three
possibilities.
 Consistent Systems (one solution)
 Inconsistent Systems (no solutions)
 Dependent Systems (Infinite number of solutions)
Consistent Systems
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
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The lines will intersect.
The point where the lines
intersect is your solution.
The solution of this graph is (1, 2)
(1,2)
Inconsistent Systems
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
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These lines never intersect as
the lines are parallel!
Since the lines never cross,
there is
NO SOLUTION!
Parallel lines have the same
slope with different yintercepts.
2
=2
1
y-intercept = 2
Slope =
y-intercept = -1
Dependent Systems
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These lines are the same!
Since the lines are on top of
each other, there are
INFINITELY MANY
SOLUTIONS!
Coinciding lines have the
same slope and
y-intercepts.
2
=2
1
y-intercept = -1
Slope =
What is the solution of the system graphed
below?
1.
2.
3.
4.
(2, -2)
(-2, 2)
No solution
Infinitely many solutions
1) Find the solution to the following system by graphing:
2x + y = 4
x-y=2
Graph both equations.
I will graph using x- and y-intercepts (plug in zeros), but
you can also rewrite in to y =mx + b form.
2x + y = 4
(0, 4) and (2, 0)
x–y=2
(0, -2) and (2, 0)
Graph the ordered pairs.
Graph the equations.
2x + y = 4
(0, 4) and (2, 0)
x-y=2
(0, -2) and (2, 0)
Where do the lines intersect?
(2, 0)
Check your answer!
To check your answer, plug the
point back into both
equations.
2x + y = 4
2(2) + (0) = 4
x-y=2
(2) – (0) = 2
Nice job…let’s try another!
2) Find the solution to the following system:
y = 2x – 3
-2x + y = 1
Graph both equations. Put both equations in
slope-intercept or standard form. I’ll do slopeintercept form on this one!
y = 2x – 3
y = 2x + 1
Graph using slope and y-intercept
Graph the equations.
y = 2x – 3
m = 2 and b = -3
y = 2x + 1
m = 2 and b = 1
Where do the lines intersect?
No solution!
Notice that the slopes are the same with different
y-intercepts. If you recognize this early, you don’t
have to graph them!
Check your answer!
Not a lot to check…Just make
sure you set up your equations
correctly.
I double-checked it and I did it
right…
What is the solution of this system?
3x – y = 8
2y = 6x -16
1.
2.
3.
4.
(3, 1)
(4, 4)
No solution
Infinitely many solutions
Solving a system of equations by Graphing
Let's summarize! There are 3 steps to solving a
system using a graph.
Step 1: Graph both equations.
Graph using slope and y – intercept
or x- and y-intercepts. Be sure to use
a ruler and graph paper!
Step 2: Do the graphs intersect?
This is the solution! LABEL the
solution!
Step 3: Check your solution.
Substitute the x and y values into
both equations to verify the point is a
solution to both equations.
Solving Systems of Equations
using Substitution
Steps:
1. Solve one equation for one variable (y= ; x= ; a=)
2. Substitute the expression from step one into the
other equation.
3. Simplify and solve the equation.
4. Substitute back into either original equation to find
the value of the other variable.
5. Check the solution in both equations of the system.
Example #1:
y = 4x
3x + y = -21
Step 1: Solve one equation for one variable.
y = 4x
(This equation is already solved for y.)
Step 2: Substitute the expression from step one into
the other equation.
3x + y = -21
3x + 4x = -21
Step 3: Simplify and solve the equation.
7x = -21
x = -3
Example #1 cont:
y = 4x
3x + y = -21
Step 4: Substitute back into either original
equation to find the value of the other
variable.
3x + y = -21
3(-3) + y = -21
-9 + y = -21
y = -12
Solution to the system is (-3, -12).
y = 4x
3x + y = -21
Step 5: Check the solution in both equations.
Solution to the system is (-3,-12).
y = 4x
-12 = 4(-3)
-12 = -12
3x + y = -21
3(-3) + (-12) = -21
-9 + (-12) = -21
-21= -21
Example #2:
x + y = 10
5x – y = 2
Step 1: Solve one equation for one variable.
x + y = 10
y = -x +10
Step 2: Substitute the expression from step one into
the other equation.
5x - y = 2
5x -(-x +10) = 2
Example #2 cont:
x + y = 10
5x – y = 2
Step 3: Simplify and solve the equation.
5x -(-x + 10) = 2
5x + x -10 = 2
6x -10 = 2
6x = 12
x=2
Example #2 cont:
x + y = 10
5x – y = 2
Step 4: Substitute back into either original
equation to find the value of the other
variable.
x + y = 10
2 + y = 10
y=8
Solution to the system is (2,8).
Example #2 cont:
x + y = 10
5x – y = 2
Step 5: Check the solution in both equations.
Solution to the system is (2, 8).
x + y =10
2 + 8 =10
10 =10
5x – y = 2
5(2) - (8) = 2
10 – 8 = 2
2=2
Solving Systems of Equations using Elimination
(also called Solving by using Addition)
Steps:
1. Place both equations in Standard Form, Ax + By = C.
2. Determine which variable to eliminate with Addition or
Subtraction.
3. Solve for the variable left.
4. Go back and use the found variable in step 3 to find
second variable.
5. Check the solution in both equations of the system.
EXAMPLE #1:
5x + 3y = 11
5x = 2y + 1
STEP 1:
Write both equations in Ax + By = C form.
5x + 3y =1
5x - 2y =1
STEP 2:
Multiply the 2nd equation by -1.
5x + 3y =11
-5x + 2y =1
STEP 3:
Add like terms and solve.
5x + 3y =11
-5x + 2y = -1
5y =10
y=2
5x + 3y = 11
5x = 2y + 1
STEP 4:
Solve for the other variable by substituting
into either equation.
5x + 3y =11
5x + 3(2) =11
5x + 6 =11
5x = 5
x=1
The solution to the system is (1,2).
5x + 3y= 11
5x = 2y + 1
Step 5: Check the solution in both equations.
The solution to the system is (1,2).
5x + 3y = 11
5(1) + 3(2) =11
5 + 6 =11
11=11
5x = 2y + 1
5(1) = 2(2) + 1
5=4+1
5=5
Solving Systems of Equations
using Elimination
Steps:
1. Place both equations in Standard Form, Ax + By = C.
2. Determine which variable to eliminate with Addition or
Subtraction.
3. Solve for the remaining variable.
4. Go back and use the variable found in step 3 to find the second
variable.
5. Check the solution in both equations of the system.
Example #2:
x + y = 10
5x – y = 2
Step 1: The equations are already in standard
form:
x + y = 10
5x – y = 2
Step 2: Adding the equations will eliminate y.
x + y = 10
x + y = 10
+(5x – y = 2)
+5x – y = +2
Step 3:
Solve for the variable.
x + y = 10
+5x – y = +2
6x = 12
x=2
x + y = 10
5x – y = 2
Step 4:
Solve for the other variable by
substituting into either equation.
x + y = 10
2 + y = 10
y=8
Solution to the system is (2,8).
x + y = 10
5x – y = 2
Step 5: Check the solution in both equations.
Solution to the system is (2,8).
x + y =10
2 + 8 =10
10=10
5x – y =2
5(2) - (8) =2
10 – 8 =2
2=2
Using Elimination to Solve a Word
Problem:
Two angles are supplementary. The
measure of one angle is 10 degrees
more than three times the other. Find
the measure of each angle.
Using Elimination to Solve a Word
Problem:
Two angles are supplementary. The
measure of one angle is 10 more
than three times the other. Find the
measure of each angle.
x = degree measure of angle #1
y = degree measure of angle #2
Therefore x + y = 180
Using Elimination to Solve a Word
Problem:
Two angles are supplementary. The
measure of one angle is 10 more
than three times the other. Find the
measure of each angle.
x + y = 180
x =10 + 3y
Using Elimination to Solve a Word
Problem, cont:
x + y = 180
x =10 + 3y
x + y = 180
x - 3y = 10
I will multiply the second equation by -1 then add like terms.
x + y = 180
-x + 3y = -10
4y =170
y = 42.5
Using Elimination to Solve a Word
Problem:
Substitute the 42.5 to find the other angle.
x + 42.5 = 180
x = 180 - 42.5
x = 137.5
(137.5, 42.5)
Using Elimination to Solve a Word
problem:
The sum of two numbers is 70
and their difference is 24. Find
the two numbers.
x = first number
y = second number
Therefore, x + y = 70
Using Elimination to Solve a Word
Problem:
The sum of two numbers is 70
and their difference is 24. Find
the two numbers.
x + y = 70
x – y = 24
Using Elimination to Solve a Word
Problem:
x + y =70
x - y = 24
2x = 94
x = 47
47 + y = 70
y = 70 – 47
y = 23
(47, 23)