4.4 Convert Linear Equations From Standard Form - Zapp's Math

4.4 Convert Linear Equations From Standard Form

Investigate

standard form

?a linear equation

of the form Ax + By + C = 0

?A is a whole number ?B and C are integers ?A and B cannot both

equal zero

Many businesses rely on information obtained from linear relations. Profit, loss, and break-even points are analysed in an effort to run successful businesses. Sometimes, it is useful to present a linear relation in a different form. In this section, you will explore how different representations of linear equations are related.

Find the Initial Value and the Rate of Change

The equation 13x - 2y + 24 = 0 is written in standard form . In this equation, y represents the total cost of a school field trip and x represents the total number of students going on the trip.

1. Use the graph. a) Find the slope of the line. What does the slope represent in this situation? b) What is the y-intercept? What does this value represent? c) Write the equation of the line in the form y = mx + b.

Z YZ

2. Rearrange the equation 13x ? 2y + 24 = 0 to isolate y. How does the result compare to your answer to part c)?

Y

3. Reflect Explain why it is sometimes useful to rearrange an equation into slope y-intercept form.

184 MHR ? Chapter 4

Example

1 Rewrite a Linear Equation in Slope y-Intercept Form

Rearrange the equation 3x + y - 5 = 0 into slope y-intercept form, then identify the slope and the y-intercept.

Solution To write the equation in slope y-intercept form, rearrange the equation to isolate y.

3x + y - 5 = 0 3x + y - 5 + 5 = 0

3x + y = 5 3x + y - 3x = 5 - 3x

y = 5 - 3x or y = -3x + 5 The slope of this linear relation is -3 and the y-intercept is 5.

Example

2 Rearrange a Linear Equation Involving Fractions

Rewrite the equation 2x + 3y - 9 = 0 in slope y-intercept form, then state the slope and y-intercept.

Solution

2x + 3y - 9 = 0

2x + 3y - 9 - 2x = 0 - 2x

3y - 9 + 9 = -2x + 9

3y = -2x + 9 Divide each term in the equation by 3.

_3y

3

=

-_23x

+

_ 9

3

The

slope

is

-_23 ,

y = -_23x + 3

and the y-intercept

is

3.

4.4 Convert Linear Equations From Standard Form ? MHR 185

Example

3 Find the Number of Tickets Needed to Be Sold

Tickets for a local theatre production are $2 per adult and $1 per child. The total revenue, the amount of money the theatre made, R, is 2x + y = R where x represents the number of adult tickets sold, and y represents the number of children's tickets sold.

The theatre company wants to bring in revenue of $750 for each of the next three shows. The number of adult tickets sold for these shows is 200, 225, and 175 respectively.

a) Write the revenue equation in slope y-intercept form. b) How many children's tickets must be sold for each show to achieve

the revenue target?

c) What is the advantage of rearranging the equation first?

Solution

a)

2x + y = R

The total revenue wanted is $750.

2x + y = 750

2x + y - 2x = 750 - 2x

y = -2x + 750

b) First show:

Second show:

Third show:

Substitute x = 200. Substitute x = 225. Substitute x = 175.

y = -2(200) + 750 y = -2(225) + 750 y = -2(175) + 750

= -400 + 750 = 350

= -450 + 750 = 300

= -350 + 750 = 400

To meet the revenue target, 350 children's tickets must be sold for the first show, 300 for the second show, and 400 for the third show.

c) You could substitute each x-value into 2x + y = 750, but each time, you would have to rearrange the equation to find the answer. Rearranging the equation into slope y-intercept form allows you to substitute and evaluate without rearranging.

186 MHR ? Chapter 4

Key Concepts

? A linear equation can be represented in different ways. ? To write a linear equation in slope y-intercept form, rearrange the

equation to isolate y.

Discuss the Concepts D1. Write an example of a linear equation in standard form, and an

example of a linear equation in slope y-intercept form.

D2. Marc states that the equations y = -_13x + 5 and x + 3y - 15 = 0

represent the same linear relation. Is he correct? Explain.

Practise the Concepts A

1. Find the slope and y-intercept of each linear relation, then write the equation for the relation in slope y-intercept form.

a)

Z

b)

Z

Y YZ

YZ Y

c)

Z

Y YZ

d)

Z

Y YZ

4.4 Convert Linear Equations From Standard Form ? MHR 187

For help with question 2, refer to Example 1.

2. Rewrite each equation in slope y-intercept form.

a) 2x + y - 1 = 0

b) 3x - y - 5 = 0

c) 2x + y - 4 = 0

d) 5x + y + 8 = 0

e) x - y + 1 = 0

f) 2x - y - 3 = 0

For help with question 3, refer to Example 2.

3. Rewrite each equation in slope y-intercept form, then state the

slope and the y-intercept.

a) 2x - y + 4 = 0

b) 3x + y - 2 = 0

c) x - y + 4 = 0

d) 3x + y + 11 = 0

e) 8x - y - 5 = 0

f) 2x + y + 7 = 0

4. Rewrite each equation in slope y-intercept form. State the

slope and the y-intercept of each.

a) 5x - 5y - 15 = 0

b) 2x - 3y + 12 = 0

c) 8x + 4y - 20 = 0

d) x - 2y + 10 = 0

e) x - 5y + 15 = 0

f) 3x - 4y + 12 = 0

g) 8x - 6y - 36 = 0

h) 3x + 6y + 18 = 0

Apply the Concepts B

For help with question 5, refer to Example 3.

Math Connect

5. A sightseeing train runs tours at four different times on Saturdays.

A 5-star sightseeing train will soon be running on the new Qinghai-Tibet railway in China.

An adult ticket is $3 and a child's ticket is $1. One Saturday, the total ticket revenue was $750. On this day, 150 tickets were sold for the first tour, 95 for the second, 125 for the third, and 96 for the fourth.

The transparent cars will allow views on all sides. Passengers will be able to shower

a) Write an equation to model the total revenue for this Saturday. b) Rearrange the equation to isolate the variable representing

children's tickets. c) Find the total number of children's tickets sold on this Saturday.

on the train and

enjoy dance

performances

and karaoke. A

sightseeing holiday

on this train will

cost over $1000

per day.

188 MHR ? Chapter 4

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