Chapter 3
Section 2: Solving Linear
Systems Algebraically
Comb
Sub
THE SUBSTITUTION METHOD
3x + 4y = -10
3x + y = 2
STEP 1: Solve one
equation for one of
its variables.
STEP 2:
Substitute the
expression from
Step 1 into the
other equation
and solve for the
other variable
STEP 3:
Substitute the
value from Step
2 into the
revised equation
from Step 1 and
solve.
3x + y = 2
y=2-3x
-3x + 2
y=-3x
3x + 4(
) = -10
3x – 12x + 8 = -10
-9x=-18
2
x=2
y=2-3( )
y=2-6
y=-4
THE LINEAR
COMBINATION METHOD
STEP 1: Multiply one or
both of the equations by
a constant to obtain
coefficients that differ
only in sign for one of the
variables.
STEP 2: Add the revised
equations from Step 1.
Combining like terms will
eliminate one of the
variables. Solve for the
remaining variable
STEP 3: Substitute the
value obtained in Step 2
into either of the original
equations and solve
for the other variable..
special
4(3x – 6y = 9 )
3(-4x + 7y = -16 )
12x – 24y = 36
-12x + 21y = -48
-3y = -12
y= 4
3x- 6(4) = 9
3x-24 = 9
3x=33
-4x + 7(4) = -16
-4x +28 = -16
-4x=-44
LINEAR SYSTEMS WITH
MANY OR NO SOLUTIONS
Remember when graphing two lines could
One Solution
– Intersect
No Solutions
– Be parallel
– Be the same line Infinite Number of
Solutions
Each of these told us something about the
number of solutions for the system of
equations
When solving analytically, what do solutions
look like?
(x, y)
One Solution
Impossible Statement
No Solutions
3=0
Infinite Number Always true statement
0=0
of Solutions
Solve the linear system.
-2( x - 2y = 3 )
(2x - 4y = 7 )
-2x + 4y = 6
2x - 4y = 7
0= 13
This statement is impossible.
There is no solution.
Solve the linear system.
15( 6x – 10y = 12 )
6( -15x + 25y = -30 )
75x - 150y = 180
-75x - 150y =-180
0= 0
This statement is always true.
There are an infinite number of
solutions.
• CATERING: a caterer is planning a
party for 64 people. The customer
has $150 to spend. A $39 pan of
pasta feeds 14 people and a $12
sandwich tray feeds 6 people. How
many pans of pasta and how many
sandwich trays should the cater
make?
• To raise money for a new football
uniforms, your school sells silkscreened T-shirts. Short sleeve Tshirts cost the school $5 each and
are sold for $8 each. Long sleeve Tshirts cost the school $7 each and
are sold for $12 each. The school
spends a total $2500 on T-shirts and
sells all of them for $4200. How
many of the short sleeve T-shirts are
sold?