Download Section 4.2 Solving Systems of Linear Equations by the Substitution

Math 123 - Section 4.2 - Solving Systems of Linear Equations by the Substitution Method - Page 1
Section 4.2
Solving Systems of Linear Equations by the Substitution Method
I. Procedure
A. Choose a variable to solve for in one of the equations.
B. Substitute for the solved variable in the other equation.
C. Simplify.
D. Solve the new equation.
E. Substitute the number you found into the other equation to find the unsolved for
variable.
F. Check your solutions in both equations.
G. Write your solution as an ordered pair (x, y).
II. Types of Systems
A. A system is consistent and independent if it has exactly one solution point (x, y).
B. A system is inconsistent if it has no solution – the null set.
C. A system is dependent if it has infinitely many solutions.
III. Examples
A. Remember that after you have substituted and simplified, if the variable does not
disappear, you have a consistent and independent system.
B. If after you have substituted and simplified, the variable disappears, then
1. If you get a true statement, you have a dependent system.
2. If you get a false statement, you have an inconsistent system.
C. Examples – Solve the following systems of equations.
y  x  2
2 y   x  2
1. 
First, we notice that in the first equation, “y” is already solved for. So
substitute x + 2 in for y in the second equation to get:
2(x + 2) = x – 2
Let’s simplify this to get:
2x + 4 = x – 2
Let’s begin solving by adding x to both sides to get:
3x + 4 = 2
Now subtract 4 from both sides to get:
3x = 6
Divide by 3 to get:
x = 2
Now we will find y by using the first equation:
y = 2 + 2
OR
y=0
So it appears that our solution is the ordered pair (2, 0). Let’s check
this in both equations to verify this:
0 = 2 + 2 OR 0 = 0 True
2(0) =  (2) – 2 OR 0 = 2 – 2 OR 0 = 0 True
Answer: (2, 0)
© Copyright 2012 by John Fetcho. All rights reserved
Math 123 - Section 4.2 - Solving Systems of Linear Equations by the Substitution Method - Page 2
4 x  y  11
2. 
 2x  3y  5
First, we have to solve one of the equations for one of the variables. I
choose to solve the first equation for y. So I need to add 4x to both sides
to get:
y = 4x – 11
Next, we substitute this into the second equation for y and solve.
2x – 3(4x – 11) = 5
Now distribute 3 to get:
2x – 12x + 33 = 5
Combine like terms to simplify.
–10x + 33 = 5
Subtract 33 from both sides of the equation.
–10x = –28
Divide both sides of the equation by –10 to isolate x.
28 14
14

Now, substitute
in for x to get:
x=
10 5
5
56
1
 14 
Write the solution as an ordered pair.
y  4    11 
 11 
5
5
5
 
Answer:
 14 1 
 5 ,5 


1

y 
x  2
3
 x  3 y  6
3. Now you try one: 
Answer: Dependent System, {(x, y) | y =
x

4.  6
 x
y
2
 2y

1
x  2}
3
1
3
 3

We don't like fractions, so let's multiply the first equation by the LCD, 6.
 x
y
1
6   6   6 
 6
 3
2
 x

2y
 3

This will leave us with:
 x  3y  2

x  2y   3
Let's solve the first equation for x by adding 3y to
both sides.
x – 3y + 3y = 2 + 3y
x = 2 + 3y
2 + 3y + 2y = 3
2 + 5y = 3
2 – 2 + 5y = 3 – 2
5y = 5
y = 1
x = 2 + 3(1)
x = 2 – 3 = 1
Now let's substitute into the second equation.
Simplify the left-hand side.
Subtract 2 from each side.
Simplify both sides.
Divide both sides by 5.
Substitute to find x.
Simplify.
Write the solution point.
Answer: (1, 1)
© Copyright 2012 by John Fetcho. All rights reserved
Math 123 - Section 4.2 - Solving Systems of Linear Equations by the Substitution Method - Page 3
5. The sum of two numbers is 62. One number is 12 more than the other. Find the
numbers. Page 295, #34
Let x = the first number
Let y = the second number
"sum" means add. So the sum of two numbers is 62 means
x + y = 62
"more than" means add. "is" means equals.
x = 12 + y
Substitute into the first equation for x.
12 + y + y = 62
Simplify the left-hand side.
12 + 2y = 62
Subtract 12 from both sides.
12 – 12 + 2y = 62 – 12 Simplify.
2y = 50
Divide both sides by 2.
y = 25
Substitute into the second equation to find x.
x = 12 + 25 = 37
Answer the question.
Answer: The numbers are 25 and 37.
6. The music business is evolving into a digital marketplace. The bar graph (middle
of left-hand column, page 295) shows that from 2004 through 2009, CD album
sales declined, while digital track sales grew. The data can be described by the
following models:
Album Sales: y = 95x + 770
Track Sales: y = 219x + 142
where x is the number of years after 2004 and y is sales in millions.
a. Solve the system. Page 295, #42
Since both equations have been solved for y, we set the two equations equal to
each other.
95x + 770 = 219x + 142
95x + 95x + 770 = 219x + 95x + 142
770 – 142 = 314x + 142 – 142
628 = 314x
2=x
Add 95x to both sides.
Subtract 142 from both sides.
Simplify.
Divide both sides by 314
Chose one of the equations to substitute for x
and find y.
y = 95(2) + 770 = 190 + 770 = 580
OR
y = 219(2) + 142 = 438 + 142 = 580 Write the solution point.
Answer: (2, 580)
By the way, we would interpret this as the following:
In 2006 (2004 + 2), album sales and track sales will be the same, 580 million each.
© Copyright 2012 by John Fetcho. All rights reserved