This document provides instruction on solving systems of linear equations and inequalities. It introduces systems of equations, discusses three methods for solving them (graphing, elimination, and substitution), and provides examples of each method. It also assigns practice problems for students to work on solving systems by graphing. Finally, it previews content on inequalities that will be covered tomorrow and assigns a test on coordinate planes for the next class.

1.  Begin Unit On Systems of
Equations/Inequalities
 Practice:
a. Together
b. On your own
 Khan Academy Topics Posted & Due
by the 17th.:
Today:

2. I Happened to notice
this last night.....

6. Systems of Equations
 To solve for one variable, only one equation is needed.
 When solving for two variables, two or more equations
are required to know what the solutions may be.
 Three unknowns require three equations, ....
 These are systems of equations.

7. SYSTEMS OF LINEAR EQUATIONS
So far, we have solved equations with one variable: 3x + 5 = 35
and found combinations of solutions in two variables.
Now we will solve multiple equations at the same
time, looking for an ordered pair which solves each
equation, and thus is a solution for both.
Example:
3x + 3y = -3
y = x + 1
We’ll take a quick look at all of the methods before focusing
on each one separately.
3x + 5y = 35
There are 3 methods for solving systems of equations:
1) By Graphing 2) By Elimination 3) By Substitution

8. Example:
3x + 3y = -3
y = x + 1
SOLVING SYSTEMS BY GRAPHING:
x
y
1. Write each equation into slope-intercept form.
2. Graph both equations in
the same coordinate plane.
3. Find the point of intersection.
4. Check your answer.
- Plug that point into both equations
and make sure that it is true for both.
y mx b 

9. Example:
3x + 3y = -3
y = x + 1
SOLVING SYSTEMS BY ELIMINATION:
1. Arrange the like variables in columns.
2. Pick a variable, x or y, and make the two
equations opposites using multiplication.
3. Add the equations together (eliminating a
variable) and solve for the remaining variable.
4. Substitute the answer into one of the
ORIGINAL equations and solve.
5. Check your solution.

10. SOLVING SYSTEMS BY SUBSTITUTION:
Example:
3x + 3y = -3
y = x + 1
1. Solve one of the equations for x or y.
2. Substitute your new expression from Step
1 into the other equation and solve for the
variable.
3. Plug that solved variable into the other equation
from Step 1 and solve for the other variable.
4. Check your answers by plugging it into the
original equations.
- Get x or y by itself.

12. Solving Linear Systems by graphing
x
y
Consider the following system: x – y = –1
x + 2y = 5
Using the graph to the right, we
can see that any of these ordered
pairs will make the first equation
true since they lie on the line.
Notice: Any of these points will
make the second equation true.
However, there is ONE point that
makes both true together…
(1, 2)
The point where they intersect makes both equations true
at the same time, and is the solution to this system
Graph this line
Then this line
Plug the coordinates into both equations to check if each
equations is true.

13. SOLUTIONS TO SYSTEMS OF LINEAR EQUATIONS
If the system of linear equations is going to have a solution,
then the solution will be an ordered pair (x , y) where x and y
make both equations true at the same time.
These are the three possible solutions:

14. Practice: 1 of 3
x
y
Rewrite the two equations in slope-
intercept form:
Plot points for each line.
Draw the lines.
These two equations represent the same line.
Therefore, this system of equations has infinitely many solutions.
Solve by Graphing

15. The two equations
in slope-intercept
form are:
x
y
Plot points for each line.
Draw in the lines.
This system of equations represents two parallel lines.
This system of equations has no solution because these
two lines have no points in common.
Practice: 2 of 3

16. x
y
The two equations in
slope-intercept form are:
Plot points for each line.
Draw in the lines.
This system of equations represents two intersecting lines.
The solution to this system of equations is a single point (3,0)
Practice: 3 of 3

17. SOLVING SYSTEMS BY ELIMINATION:(3)
1. Arrange the like variables in columns.
2. Pick a variable, x or y, and make the two
equations opposites using multiplication.
3. Add the equations together (eliminating a
variable) and solve for the remaining variable.
4. Substitute the answer into one of the
ORIGINAL equations and solve.
5. Check your solution.

18. Tomorrow....
a. Solving by Substitution
Right Now......
Class Work 3.3

21. Rewrite the inequality 4x < -
𝟐
𝟑
y +
𝟏
𝟐
in standard form
with integer values only.
24x + 4y < + 3
Inequalities
4x +
𝟏
𝟑
y < -
𝟏
𝟐
5 min. break, then the last test
of the coordinate plane unit.
Solving Systems of Equations
Tomorrow:
V.3 # 2
V.4 # 4; E. None