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# Module 5

By Joel Carroll,2014-05-07 13:52
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Module 5

Module Five:

Miscellaneous Topics

In this module, we will review radical expressions, graphing linear equations, and

solving systems of equations.

Example 1.

65 Simplify:350xy

Method A.

643252xyyStep 1.

32Step 2. 3?5?x?y2y

32Step 3. 15xy2y

Method B.

22222325?2?x?x?x?y?y?yStep 1.

Step 2. 3?5?x?x?x?y?y?2y

32Step 3. 15xy2y

Example 2.

2 Simplify: 5a3b

????5a3b5a3bStep 1.

5a?5a3b?3bStep 2.

22Step 3. 25a9b2Step 4. 25a ? 3b

275abStep 5.

Example 3.

Perform the indicated operations:

518262

59?2?2?62Step 1.

Step 2. 152?2?62

??15?1?62Step 3.

Step 4. 202

Interpreting Graphs of Linear Equations

The next topic is interpreting the graph of a linear equation. In these examples, a graph

will be given. We will ask questions whose answers require interpretation of the graph.

Example 4.

The graph of y + 2x = 4 is shown below.

y

4

2x

– 424– 2

– 2

– 4

A. What are the coordinates of the x-intercept?

Step 1. x = 2 and y = 0

Step 2. (2, 0)

B. What are the coordinates of the y-intercept?

Step 1. x = 0 and y = 4

Step 2. (0, 4)

C. Does the line contain the origin?

Method A.

Step 1. The origin is the point (0, 0).

Step 2. Using substitution where x = 0 and y = 0 we obtain:

y + 2x = 4

0 + 2 ? 0 = 4

0 = 4

Step 3. The line does not contain the origin.

Method B.

Step 1. Visual inspection of the graph. The line does not pass

through the point (0, 0). Therefore, the line does not contain

the origin.

D. Determine whether the slope of the line is positive, negative, or

neither.

Method A.

Step 1. In the formula, y = mx + b, the slope is represented by the

letter m.

Step 2. Solving y + 2x = 4 for y, we obtain y = 2x + 4 which is in

the form y = mx + b.

Step 3. m = 2 and the slope is negative.

Method B.

Step 1. By visual inspection of the graph, as we view the line from

left to right, the y-coordinates decrease in value.

Step 2. The slope is negative.

Example 5.

Which of the following is the graph of 3x y = 0?

AByy..

xx

Step 1. Solving 3x y = 0 for y, we obtain y = 3x which is in the form

y = mx + b.

Step 2. The slope is m = 3, and the slope is positive.

Step 3. Since the slope is positive, graph A is the answer..

Systems of Equations

The third topic we will discuss is solving systems of equations

Example 6.

Solve the following system of equations for x and y using the

substitution method.

4x y = 22

x = 3y

Step 1. Substitute 3y for x in 4x y = 22.

4(3y) y = 22

Step 2. Solve for y:

12y y = 22

11y = 22

y = 2

Step 3. Determine the value of x by substituting y = 2 into x = 3y.

x = 3y

x = 3(2)

x = 6

Step 4. x = 6 , y = 2 written as (6, 2)

Example 7.

Solve the following system of equations for x and y using the

elimination method.

3x + y = 6

2x 4 = y

Step 1. Rewrite 2x 4 = y as 2x y = 4

Step 2. Add 2x y = 4 and 3x + y = 6

2x y = 4

+ 3x + y = 6

5x = 10

Step 3. Solve for x.

5x = 10

x = 2

Step 4. Determine the value of y by substituting x = 2 into the

equation 3x + y = 6.

3x + y = 6

3(2) + y = 6

6 + y = 6

y = 0

Step 5. x = 2, y = 0 written as (2, 0)

Example 8.

Solve the following system of equations for x and y using the

elimination method.

x + 11 = 7y

x 7y = 9

Step 1. Rewrite x + 11 = 7y as x 7y = 11

Step 2. Subtract x 7y = 9 from x 7y = 11

x 7y = 11

( x 7y) = (9)

0 = 20

Step 3. Since 0 = 20 is a false statement, the system has

no solution and is called an inconsistent system.

Practice problems.

62 1.Simplify:320xyz

Solution to problem 1:

62334?5xyz?3?2?x?z5y

3 ?6xz5y

234 2.Simplify:2k5mn

Solution to problem 2:

234 4k(5mn)

234 ?20kmn

3.Simplify:51282748

Solution to problem 3:

54?3?89?3?16?3 ?5?23?8?33?43

?103?243?43

??103

4. The graph of 5x 4y = 20 is shown below.

a) What are the coordinates of the x-intercept?

b) What are the coordinates of the y-intercept?

c) Does the line contain the origin?

d) Determine whether the slope of the line is positive, negative or neither.

y (5 3/5, 2)

x

(4,0)

(0,-5)

Solution to problem 4:

a) (4, 0)

b) (0, 5)

c) no

5 d)positive;theslopeis4

5. Which of the following is the graph of x + 2y = 0?

yyA

B

xx

Solution to problem 5:

A

6. Solve the following system of equations for x and y using the substitution

method.

x 3y = 1 and 2x + 8 =y

Solution to problem 6:

Solve x 3y = 1 for x and obtain

x = 1 + 3y

Then substitute into 2x + 8 = y:

2(1 + 3y) + 8 = y

Now solve for y.

2 + 6y + 8 = y

10 + 6y = y

10 = 5y

2 = y

Substitute - 2 for y into x = 1 + 3y

and obtain

x = 1 + 3( 2)

x = 5

Hence the solution is the ordered pair ( 5, 2)

7. Solve the following system of equations for x and y using the elimination

method.

x + 4y = 16 and 6x + y = 27

Solution to problem 7:

Multiply the first equation by 6

6(x + 4y) = 6 (16) resulting in 6x 24y = 96

6x 24y = 96

6x + y = 27

23y = 69

y = 3

Substituting 3 for y into x + 4y = 16 we obtain

x + 4(3) = 16

x + 12 = 16

x = 4

the ordered pair is (4, 3)

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