The document discusses different methods for solving systems of linear equations: 1) Graphing both equations on the same coordinate plane and finding the point of intersection. 2) Using the substitution method, which involves replacing one variable with an equivalent expression containing the other variable. 3) Using the elimination method, which involves adding or subtracting equations to eliminate a variable and then solving for the remaining variable. The document provides examples of using substitution and elimination methods and discusses the possible solutions when graphing systems of linear equations.

1. Systems of Equations
Presented by:
Cara Varon
EDUU 551 Applications of Computers
Brandman University

2. Activating Schema
• Chapters 2 and 3: You learned to solve
multi-step problems involving linear
equations and inequalities in one variable
and provided justification for each step.
• Chapter 5: You learned how to graph a
linear equation and compute the x- and y-
intercepts
• Chapter 5: You learned to verify that a
point lies on a line, given the equation of
the line.

3. Vocabulary
Definitions
1. A system of linear equations is a
situation in which two or more linear
equations are together in the same
scenario
2. A solution of the system of linear
equations is an ordered pair in a system
that makes all of the equations true.

4. Solving Systems by Graphing
Steps to follow:
1. Graph both
equations on the
same coordinate
plane
2. Find the point of
intersection
3. Check to see if
the point of
intersection
makes both
equations true

5. Three Possible Solutions of Systems of Linear
Equations
• If the lines have
different slopes, then
the lines intersect, so
there is only one
solution
• If the lines have the
same slope and
different y-
intercepts, then the
lines are parallel and
there are no solutions
• If the lines have the
same slope and the
same y-intercept, then
the lines are the same,
so there are infinite
solutions

6. Solving Systems Using Substitution
• Substitution Method: Another method for
solving systems of equations by replacing one
variable with an equivalent expression containing
the other variable.
• Steps:
1. Write an equation containing only one variable, and solve it.
2. Solve for the other variable in either equation.
3. The solution will be an ordered pair.
4. Check to see if the ordered pair makes both equations true.
**See example on next slide.

7. Example of Substitution Method
Problem: y=-4x+8
y=x+7
Step 1.
Start with one equation. y = -4x+8
Substitute x+7 for y. y+7 = -4x+8
Solve using Equality Properties. x = 0.2
Step 2.
Substitute 0.2 for x in y=x+7. y = -4(0.2)+8
Simplify. y = 7.2
Solution is (0.2 , 7.2)
Step 3. Check
Replace the x and y variable with the solution set.
7.2 - -4 (0.2) + 8
7.2 = 7.2

8. Solving Using Elimination Method
• Elimination Method: Another where you can
use the properties of equality to solve a
system. You can add of subtract equations to
eliminate a variable.
• Steps:
1. Look for coefficients that are opposites or each other. If there
aren’t any, you may need to multiply one or both equations by a
nonzero number. This will to produce coefficients that are
opposites of each other.
2. Eliminate one variable.
3. Solve for the remaining variable.
4. Solve for the eliminated variable using the original equations.
5. The solution is an ordered pair.
6. Check to see if the ordered pair makes both equations true.

9. Example of Elimination Method
Problem: 2x+5y = -22
10x+3y = 22
Step 1:
Eliminate one variable 5 [ 2x+5y = -22 ] 10x + 25y = -110
10x+3y = 22 10x + 3y = 22
0 + 22y = -132
Step 2:
Solve for y. 22y = -132 y = -6
Step 3:
Solve for the eliminated variable using either of the original equations.
2x + 5(-6) = -22 2x – 30 = -22 2x = 8 x=4
The solution is (4, -6 )

10. Additional Resources
If you need additional help. Please access the
following sites:
• The Khan Academy – www.khanacademy.com
• Prentice Hall Algebra 1 Textbook Homework
Video Tutor – www.PHSchool.com

11. California Content Standards –
8th Grade Algebra 1
• 9.0 Solve a system of two linear
equations in two variables algebraically
and interpret the answer graphically.
• 9.0 Solve a system of two linear
equations using three techniques:
graphing
elimination
substitution