1. TODAY:
MAKE UP TESTS?
**KHAN ACADEMY DUE TOMORROW
BEGIN SYSTEMS OF EQUATIONS

2. WARM-UP:
1.
2. What is the percent change in the price of
gas from $4.80/gal to $5.00/gal.

3. SYSTEMS OF LINEAR EQUATIONS
So far, we have solved equations with one 3x + 5 = 35
variable:two
and in
variables. That will change
In both cases, now as we solve
though we have only multiple equations
been able to solve at the same time,
one equation at a looking for an
time. ordered pair which
A system of linear equations is solves each
simply two or more linear equation, and thus
equations using the same is a solution for
variables. both.
We'll start with systems of two equations using two
variables,
then increase this to three equations and variables.

4. WHAT IS A SYSTEM 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.
If the lines are parallel, there will be no solutions. If
the equations are the same line, there will be an infinite
number of solutions.
There are several methods of solving systems of
equations; we'll look at a couple today.

5. Tell whether the ordered pair is a
solution of the given system.
Substitute 5 for x
and 2 for y in each
(5, 2); equation in the
3x – y = 13 system.
3x – y 13
0 3(5) – 2 13
2–2 0 15 – 2 13
0 0  13 13 
The ordered pair (5, 2) makes both equations true.
(5, 2) is the solution of the system.

6. Helpful Hint
If an ordered pair does not satisfy the first equation in the
system, there is no reason to check the other equations.
x + 3y = 4 Substitute –2 for
(–2, 2); –x + y = 2 x and 2 for y in
each equation in
x + 3y = 4 –x + y = 2
the system.
–2 + 3(2) 4 –(–2) + 2 2
–2 + 6 4 4 2
4 4 
The ordered pair (–2, 2) makes one equation true
but not the other.
(–2, 2) is not a solution of the system.

7. SOLVING LINEAR SYSTEMS BY GRAPHING
Consider the following system:
x – y = –1 y
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 x
(1 , 2)
Notice that line. of these points will
lie on the any
make the second equation true.
However, there is ONE point
that makes both true at the
same time…
The point where they intersect makes both
equations true at the same time.

8. Practice 1
Graph the system of equations. Determine
whether the system has one solution, no
solution, or infinitely many solutions. If the
system has one solution, determine the solution.

9. Practice 1
y The two equations in slope-intercept
form are:
x
Plot points for each line.
Draw the lines.
These two equations represent the same line.
Therefore, this system of equations has infinitely many solutions .

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

11. Practice 3
y The two equations in
slope-intercept form are:
x
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)

12. Graphing to Solve a Linear System
Let's summarize! There are 4 steps to solving a linear
system using a graph.
Step 1: Put both equations in slope - Solve both equations for y, so that
intercept form. each equation looks like
y = mx + b.
Step 2: Graph both equations on the Use the slope and y - intercept for
same coordinate plane. each equation in step 1.
Step 3: Estimate where the graphs This is the solution! LABEL the
intersect. solution!
Step 4: Check to make sure your Substitute the x and y values into both
solution makes both equations true. equations to verify the point is a
solution to both equations.

13. Like variables Solve:
must be lined
under each by ELIMINATION
other. 5x - 4y = -21
-2x + 4y = 18
We need to
eliminate
3x = -3 Divide by 3
(get rid of)
a variable. x = -1
The y’s be
will the
easiest.So,
we will add THEN----
the two
equations.

14. 5x - 4y = -21
5(-1) – 4y = -21
Substitute
your answer -5 – 4y = -21
into either 5 5
original
equation and -4y = -16
solve for the
second
variable. y=4
Answer (-1, 4)
Now check our answers
in both equations------

15. 5X - 4Y = -21
5(-1) – 4(4) = -21
-5 - 16 = -21
-21 = -21
-2x + 4y = 18
-2(-1) + 4(4) = 18
2 + 16 = 18
18 = 18

16. Like variables
Solve:
must be lined
under each
by ELIMINATION
other. 2x + 7y = 31
5x - 7y = - 45
We need to
eliminate
7x = -14 Divide by 7
(get rid of)
a variable. x = -2
The y’s will
be the
easiest. So,
we will add THEN----
the two
equations.

17. Substitute
your answer 2X + 7Y = 31
into either
original
2(-2) + 7y = 31
equation and -4 + 7y = 31
solve for the
second
4 4
variable. 7y = 35
y=5
Answer (-2, 5)
Now check our answers
in both equations------

18. 2x + 7y = 31
2(-2) + 7(5) = 31
-4 + 35 = 31
31 = 31
5x – 7y = - 45
5(-2) - 7(5) = - 45
-10 - 35 = - 45
- 45 =- 45

19. Like variables
must be lined
under each Solve:
other.
by ELIMINATION
x + y = 30
We need to eliminate
(get rid of) a variable. x + 7y = 6
To simply add this
time will not eliminate
a variable. If one of the
x’s was negative, it
would be eliminated
when we add. So we
will multiply one
equation by a – 1.

20. X + Y = 30 X + Y = 30
( X + 7Y = 6 ) -1 -X – 7Y = - 6
Now add the two
equations and
-6Y = 24
solve.
-6 -6
Y=-4
THEN----

21. Substitute
your answer X + Y = 30
into either
original
X + - 4 =30
equation and
solve for the
4 4
second
variable. X = 34
Answer (34, - 4)
Now check our answers
in both equations------

22. x + y = 30
34 + - 4 = 30
30 = 30
x + 7y = 6
34 + 7(- 4) = 6
34 - 28 = 6
6=6

23. Solve: Elimination By Multiplying
x + +y0y 4 4
0x = =
Like variables
must be lined 2x + 3y = 9
under each other.
We need to eliminate (get rid of) a variable.
To simply add this time will not eliminate a variable. If there was a –
2x in the 1st equation, the x’s would be eliminated when we add. So
we will multiply the 1st equation by a – 2.

24. ( X + Y = 4 ) -2 -2X - 2 Y = - 8
2X + 3Y = 9 2X + 3Y = 9
Now add the two
equations and
Y=1
solve.
THEN----

25. Substitute your answer into either original equation
and solve for the second variable.
X+Y=4
X +1=4
- 1 -1
X=3
Answer (3,1)
Now check our answers in both equations--

26. x+y=4
3+1=4
4=4
2x + 3y = 9
2(3) + 3(1) = 9
6+3=9
9=9

27. REVIEW: SOLVING BY GRAPHING
If the lines cross once, there will be one
solution.
If the lines are parallel, there will be no
solutions.
If the lines are the same, there will be an
infinite number of solutions.