Download 7.2 Solving Linear Systems by Substitution

Algebra
7.2
Solving Systems Using
Substitution
Solution to a System of Linear Equations
You have already learned
that the solution is the
point of intersection of
the two graphed lines.
To solve:
1.
Graph both equations
2. Identify intersection point (x,y)
3. Plug in to original equations to check
There are two algebraic methods that
allow you to solve a system easily
without graphing.
Today you will learn the method called
SUBSTITUTION.
Steps
1. In one equation, isolate one variable.
2. Substitute expression from Step 1 into
second equation and solve for the other
variable.
3. Plug in value from Step 2 into revised
equation from Step 1 and solve.
4. Check solution in both original
equations.
Solve the linear system.
3x + y = 5
2x – y = 10
y = 5 – 3x
2x – ( 5 – 3x ) = 10
Hint: It is usually
easiest to isolate
positive 1x or 1y.
y = 5 – 3(3)
2x – 5 + 3x = 10
y=5-9
5x – 5 = 10
y=-4
5x = 15
The solution is (3, - 4)
x=3
Check: 3(3) + (-4) = 5
2(3) – (-4) = 10
Solve the linear system.
2x + 6y = 15
x = 2y
2( 2y ) + 6y = 15
4y + 6y = 15
10y = 15
y = 15/10
y = 3/2
x = 2( 3/2)
x=3
The solution is (3, 3/2)
Check: 2(3) + 6(3/2) = 15
3 = 2(3/2)
You try!
x + 2y = 4
-x + y = -7
y=x–7
x + 2( x - 7 ) = 4
x + 2x - 14 = 4
3x – 14 = 4
3x = 18
x=6
y=6–7
y = -1
The solution is (6, - 1)
Check: 6 + 2(-1) = 4
-6 + (-1) = -7
Mixture Problems
Systems are often used to solved mixture
problems. These are problems when you mix
two quantities. You know the total quantity
and the total value, but not how much of
each type. To solve:
•Write one equation to describe
QUANTITY.
•Write other equation to describe
VALUE.
Set up a system and solve the
mixture problem.
An audio store sells two styles of I-pod Nanos. The 2 GB
model costs $150 and the 4GB model costs $225. Last
Saturday the store sold 22 Nanos for a total of $3900.
How many of each model did they sell?
Let x be the # of 2 GB models sold
Let y be the # of 4 GB models sold
Quantity:
Value:
x + y = 22
150x + 225y = 3900
Now, you solve.
There were 14 2GB Nanos and 8 4 GB Nanos sold.
Homework
pg. 408 #17 – 35 odd, 42