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Chapter 7
• Solving
systems of Linear
Equations
Definition
• A
linear system is a collection of two
problems. The work that we do will produce
a point that they share in common.
• We will solve using GRAPHING and
SUBSTITUTION.
Steps for Graphing
• Step
1: separate the two equations
• Step 2: turn both problems into “y =.” The
first step is to always move x. The second
step is to divide if there is a number on y.
This does not always happen.
Steps for Graphing Cont
• Step
3: collect m and b
• Step 4: reminder; always use B first. It is always
located above or below the origin. From that point
you apply Rise/Run, which is M. Rise means up if
you have a positive number and down if its negative.
Run means right if it’s positive and left if it’s
negative.
• Step 5: Look for the point where the two lines meet
and that is your answer. Label the point.
Examples
• 6x
- 3y = -15
• 2x + y = -3
STEP 1
• 6x
- 3y = -15
• 2x
+ y = -3
STEP 2: GET Y BY ITSELF
STEP 3: COLLECT M AND B
• 6x
- 3y = -15
• 
-6x
•  -3y = -6x – 15
•  -3
-3 -3
• Y = 2x + 5
• B = 5
• M = 2
• 
1
• 2x
+ y = -3
• 
-2x
• 
y = -2x – 3
• B = -3
• M = -2
• 
1
STEP 4: GRAPH USING M AND B
STEP 5: LABEL AND NAME THE POINT
WHERE THE TWO LINES MEET
Steps for using Substitution
• Step
1: Pick an equation; pick the one that
has the least amount of numbers (on the
letter) or least amount of negative of signs
• Step 2: turn it into x = or y =. You will move
the x or the y. Depends on which one has a
number on it.
Steps for using Substitution Cont
• Step
3: rewrite the other equation
• Step 4: substitute for x or y. This comes
from your result of step 2. This will create an
equation with two variable but the variables
will be the same. (Either x or y)
• Step 5: solve the equation
• Step
6: rewrite the first equation you picked
and use the result of the previous step.
• Step
7: check your answer by substituting
into the two original problems.
Examples
• 6x
- 3y = -15
• 2x + y = -3
STEP 1: pick one
STEP 2: solve for x or y
• 2x
• 
• 
+ y = -3
-2x
y = -2x – 3
Step 3: rewrite the other
step 4 : substitute; y = -2x – 3
step 5 : solve
• 6x
- 3y = -15
• 6x – 3(-2x – 3)= -15
• 6x + 6x + 9 = -15
• 12x + 9 = -15
• 
-9
• 
12x = -24
• 
12
12
• 
x = -2
Step 6: plug previous answer into the
result of step 2
• y
= -2x – 3; x = -2
• Y = -2(-2) – 3
• Y = 4 – 3
• Y = 1
• We think the answer is (-2, 1)
Step 7: check so we know the answer
x = -2 and y = 1
• 6x
- 3y = -15
• 6(-2) – 3(1)
• -12 – 3
• -15
• correct
• 2x
+ y = -3
• 2(-2) + 1
• -4 + 1
• -3
• correct
Word Problems
• You
have 7 packages of paper towels. Some
packages have 3 rolls, but some have only 1
roll. There 19 rolls altogether.
• 
• 
X+y=7
1x + 3y = 19
Word Problems
• You
buy 5 pairs of socks for $19. The wool
socks cost $5 per pair and the cotton socks
cost $3 per pair.
• X
+y=5
• 3x + 5y = 19
Word Problems
• You
have only $1 bills and $5 bills in your
wallet. There are $7 bills worth a total of $19.
• X
+y=7
• X + 5y = 19
Special Systems
• 3x
+ y = -1
• -9x – 3y = 3
• x-2y
=5
• -2x+4y = 2
Special Systems
• 2x
+y=4
• 4x – 2y = 0
• x
+y=3
• 2x + 2y = 4
Special Systems
• X
+y=3
• 2x + 2y = 6
• X
+y=3
• X + 2y = 4
To graph given when given an
equation; turn it into y=mx+b
• 
Flow Chart
•  Let x jump the tracks
•  If there is a number on
y, then we will set up
three fractions and divide
or reduce
•  Collect “b” and find it
on the y-axis
•  Collect “m” and use
rise over run to get to the
next point on my line.
If the symbol is < or > we
will draw a dashed line
•  If the symbol is < or > we
will draw a normal line
•  To shade will have to use
a test point and the slope
intercept form
•  If the test result is false
shade away from the TP, if
the test is true shade to the
TP.
• 
Inequality Systems
• X
+y<3
• -x + y > 0
Examples
• X
+ 2y < 6
• -x + y < 0
Examples