part 1: writing equations in slope intercept form. (lesson ... equations/unit 3... · 1/2x + 3y = 9...

Unit 3: Writing Equations

Copyright © 2011 Karin Hutchinson – Algebra-class.com

Unit 3: Writing Equations Chapter Review

Part 1: Writing Equations in Slope Intercept Form. (Lesson 1)

1. Write an equation that represents the line on the graph.

2. Write an equation that has a slope of -3 and a y- intercept of 10.

3. Write an equation that has a slope of ½ and passes through the origin.

4. A day care center charges $9 per hour plus a $3 snack fee for a “Parents Night Out” Special.

• Write an equation that can be used to determine the price of sending a child to the day care

center for x number of hours.

• Suppose a parent decides to send their child to the day care center for 5 hours. How much will

this cost? Justify your answer.

Part 2: Writing Equations in Standard Form (Lesson 2)

5. Rewrite the following equations in standard form.

a. y = -4x – 9 b. y = 6x + 2



6. Which term in the equation does not have an “integer coefficient”?

1/2x + 3y = 9

7. Write this equation in “correct” standard form: 1/2x + 3y = 9

8. Write the following equation in standard form: 1/3x + 1/2y = 6

9. Write an equation, in standard form, for the line shown on the graph.

a. b.

Part 3: Standard Form Word Problems (Lesson 3)

10. A new 3-D movie is out and the theatre is charging $10.50 for adults and $7 for children. You

have $50 to spend on the movies. Write an equation in standard form that represents the number of

adult, a and child, c tickets that you can purchase.

• Suppose you are taking 3 adults, at most, how many children can you take to the movies?

Justify your answer mathematically.



Part 4: Writing Equations Given Slope and a Point (Lesson 4)

11. Write an equation for the line that has a slope of -3 and passes through the point (5,-7).

12. A local tax service charges $65.25 an hour plus a filing fee. A three hour session costs $208.40.

Write an equation that can be used to find the total cost for any session.

Part 5: Writing Equations Given Two Points (Lesson 5)

13. Write an equation for the line that passes through the points: (6,-2) (-4, 8)

14. In 1991,the cost of tuition for a private school was $5,000 per year. In 2012, the cost of the same

private school is $12,500. Let x = 0 represent the year 1990.

• Write an equation that could be used to predict the cost of tuition for any given year.

• Predict what the tuition will be for the year 2015.

15. In 2003, the cost of season football tickets was $2,200. In 2012, the cost of the same season

tickets is $4,550. Let x = 0 represent the year 2000.

• Write an equation that could be used to predict the cost of the tickets for any given year.

• Predict what the cost of the tickets will be in the year 2020.

Part 6: Writing Linear Equations in Point-Slope Form (Lesson 6)

16. Write an equation that has a slope of -3 and passes through the point (2,8)

17. Write an equation that passes through the points (1,9) and (3, -8). Explain the steps that you

used to solve this problem.

18. Write an equation that has a slope of 1/2 and an x-intercept of 1. Explain how you solved this

problem.

19. Write an equation that has a y-intercept of 22 and passes through the point (3, -7).

20. Write an equation that has an x-intercept of -5 and a y-intercept of 8.



21. Mei earned $38 on her investment in 5 months. She earned $62 on the same investment in 8

months.

• Write an equation that can be used to find the amount earned (y) on Mei’s invest in x number

of months.

• About how much will Mei earn after 15 months?

• About how much is Mei earning per month? Explain how you determined your answer.

Part 7: Parallel and Perpendicular Lines (Lesson 7)

22. Write an equation for a line that passes through (2,-8) and is perpendicular to a line whose slope

is 5. Explain how you determined your answer.

23. Write an equation for a line that passes through (-3,6) and is parallel to the graph of x = 5.

24. Write an equation for a line that is perpendicular to the graph of 2x – 2y = 10 and intersects the

graph at its x-intercept.

Parallel Lines: ________________________________________________________________________

Perpendicular lines: ___________________________________________________________________

____________________________________________________________________________________



Part 8: Scatter Plots and Line of Best Fit (Lesson 8)

25. The following data represents the total U.S. E-commerce sales from 2002-2010. (Statistic from

statista.com) Let x =0 represent the year 2000.

Year E-commerce Sales (in billions of dollars)

2002 72

2004 117

2006 171

2008 214

2010 228

• Using your graphing calculator, write an equation for the line of best fit for this data.

• What does the slope and y-intercept represent in the context of this problem.?

• Predict the amount of E-commerce sales in the U.S. for the year 2012. Explain how you

determined your answer.



Unit 3: Writing Equations Chapter Review – Answer Key

Part 1: Writing Equations in Slope Intercept Form. (Lesson 1)

1. Write an equation that represents the line on the graph.

2. Write an equation that has a slope of -3 and a y- intercept of 10.

Y = mx + b is the formula for writing an equation in slope intercept form. We need to know the slope (m) and

the y-intercept (b).

Slope (m) = -3 Y-intercept (b) = 10

Y = mx+b

Y = -3x+ 10

In order to write this equation, we must identify the

y-intercept and slope from the graph.

The line crosses the y-axis at y = -6, so this is the y-

intercept. From this point, to the next identifiable

point on the graph, we count up1 unit and 2 units to

the left, so the slope is -1/2.

Y = mx+ b m = -1/2 b = -6

Y = -1/2x - 6

In order to write this equation, we must identify the

y-intercept and slope from the graph.

The line crosses the y-axis at y = 2, so this is the y-

intercept. From this point, to the next identifiable

point on the graph, we count up 3 units and 1 unit to

the right, so the slope is 3 .

Y = mx+ b m = 3 b= 2

Y = 3x + 2



3. Write an equation that has a slope of ½ and passes through the origin.

4. A day care center charges $9 per hour plus a $3 snack fee for a “Parents Night Out” Special.

• Write an equation that can be used to determine the price of sending a child to the day care

center for x number of hours.

• Suppose a parent decides to send their child to the day care center for 5 hours. How much will

this cost? Justify your answer.

Part 2: Writing Equations in Standard Form (Lesson 2)

5. Rewrite the following equations in standard form.

a. y = -4x – 9 b. y = 6x + 2

The origin is (0,0). Therefore the y-intercept for this equation is 0.

Y = mx+b m = ½ b = 0

Y = 1/2x + 0 or Y = 1/2x

Slope is the rate and the key word is usually “per” So, 9 is the slope ($9 per hour)

The y-intercept is a constant. $3 is the fee for snack. This is a constant in the problem.

Y = mx+b m = 9 b = 3

Y = 9x + 3 is the equation that represents this problem.

Since x represents the number of hours, we will substitute 5 for x in the equation.

Y = 9x+3

Y = 9(5)+3

Y = 48 Sending a child to the day care for 5 hours costs $48.

Standard Form: Ax + By = C

4x + y = -4x+4x – 9 Add 4x to both sides

4x+y = -9 Simplify

The lead coefficient is positive and there are no

fractions or decimals, so this is standard form.

Standard Form: Ax + By = C

-6x+y = 6x-6x + 2 Subtract 6x from both sides

-6x + y = 2 Simplify

-1(-6x+y) =2(-1) Multiply by -1 to make the

lead coefficient positive.

6x – y = -2 Standard form



6. Which term in the equation does not have an “integer coefficient”?

1/2x + 3y = 9

8. Write the following equation in standard form: 1/3x + 1/2y = 6

An integer is a positive or negative whole number: (…-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5…)

The set of integers does not contain fractions or decimals. Therefore, ½ is not an integer.

The term that does not have an “integer coefficient” is 1/2x.

In order for this equation to be written in standard form, we must rewrite with integer coefficients. We must

get rid of the fraction, ½. In order to do this we will multiply ALL terms by 2.

2(1/2x+3y) = 9(2)

x+ 6y = 18

There are no fractions or decimals, and the lead coefficient is positive. This is now written in

standard form.

This equation has 2 fractions, with different denominators. Therefore, in order to get rid of both fractions, I

must multiply ALL terms by the lowest common multiple (LCM). The LCM is 6.

6(1/3x+ 1/2y) = 6(6)

2x+ 3y = 36 is the proper standard form equation.



9. Write an equation, in standard form, for the line shown on the graph.

a. b.

Since the slope and y-intercept are identifiable, it

would be easiest to first write the equation in slope

intercept form, then rewrite it in standard form.

Y = mx+ b m = -2 b = 5

Y = -2x+5 Equation in slope intercept form

Now rewrite in standard form:

2x+y = -2x+2x +5 Add 2x

2x+y = 5 Simplify

2x+y = 5 is the equation in standard form.

Since the slope and y-intercept are identifiable, it

would be easiest to first write the equation in slope

intercept form, then rewrite it in standard form.

Y = mx+ b m = 4/5 b = -9

Y = 4/5x - 9 Equation in slope intercept form

Now rewrite in standard form:

5y = 5(4/5x-9) Multiply by 5

5y = 4x – 45 Simplify

-4x + 5y 4x-4x-45 Subtract 4x

-4x +5y =-45 Simplify

-1(-4x+5y) = -45(-1) Multiply by -1 to make lead

coefficient positive.

4x-5y = 45 is the equation in standard form.



Part 3: Standard Form Word Problems (Lesson 3)

10. A new 3-D movie is out and the theatre is charging $10.50 for adults and $7 for children. You

have $50 to spend on the movies. Write an equation in standard form that represents the number of

adult, a and child, c tickets that you can purchase.

• Suppose you are taking 3 adults, at most, how many children can you take to the movies?

Justify your answer mathematically.

Part 4: Writing Equations Given Slope and a Point (Lesson 4)

11. Write an equation for the line that has a slope of -3 and passes through the point (5,-7).

Price for adults(# of adults) + Price per child(# of children) = Total

10.50a + 7c = 50 is the equation that represents the number of tickets that you can purchase.

If you are taking 3 adults, then you will substitute 3 for a in your equation from above. Then solve for c.

10.50a + 7c= 50 Original Equation

10.50(3) + 7c = 50 Substitute 3 for a since there are 3 adults

31.50 + 7c = 50 Simplify: 10.50(3) = 31.50

31.50 – 31.50 + 7c = 50-31.50 Subtract 31.50 from both sides

7c = 18.50 Simplify: 50-31.50 = 18.50

7c/7 = 18.50/7 Divide by 7 on both sides

C= 2.64

Since you can’t take “part” of a child to the movies, you will be able to take 2 children to the

movies if you take 3 adults.

We know: m = -3 x = 5 y = -7 we need to find b in order to write an equation in slope intercept

form. Let’s substitute what we know and solve for b.

Y = mx=b

-7 = -3(5) + b Substitute for m, x, and y

-7 = -15 + b Simplify: -3(5) = -15

-7 + 15 = -15+15+b Add 15 to both sides

8 = b The y-intercept (b) = 8

Now that we know: m = -3 and b = 8, we can write an equation in slope intercept form:

Y = mx+ b y = -3x+ 8



12. A local tax service charges $65.25 an hour plus a filing fee. A three hour session costs $208.40.

Write an equation that can be used to find the total cost for any session.

Part 5: Writing Equations Given Two Points (Lesson 5)

13. Write an equation for the line that passes through the points: (6,-2) (-4, 8)

In this problem, we know the slope is: 65.25 because this is the rate per hour. We also know that a 3 hour

session is 208.40. This is an ordered pair, because 3 hours is directly related to 208.40 (3, 208.40).

So, we know: m = 65.25 x = 3 y = 208.40 Now we need to solve for b.

Y=mx+b

208.40 = 65.25(3) + b Substitute for m, x, and y.

208.40 = 195.75+ b Simplify: 65.25(3) = 1953.75

208.40 – 195.75 = 195.75 – 195.75 + b Subtract 195.75 from both sides

12.65 = b The y-intercept is 12.65.

Now, write an equation using the slope and y-intercept.

Y =mx+b

Y = 65.25x + 12.65 This is the equation that could be used to find the total cost for any session.

Since we are given two points, we must find the slope and y-intercept. We will first find the slope using the

slope formula. Then we will use the slope and 1 point to find the y-intercept.

y2- y1 = 8-(-2) = 10 The slope is -1

x2 – x1 -4 – 6 -10

Now we will use the slope and 1 point to find the y-intercept. Let’s use (6, -2)

Y= mx+ b m = -1 x = 6 y = -2

2 = -1(6) + b Substitute for m, x, and y

2 = -6 + b Simplify: (-1)(6) = -6

2+6 = -6+6 + b Add 6 to both sides

8 = b The y-intercept = 8

Y = mx+ b m = -1 b = 8

Y = -x + 8 is the equation that passes through the points (6,-2) & (-4,8)



14. In 1991,the cost of tuition for a private school was $5,000 per year. In 2012, the cost of the same

private school is $12,500. Let x = 0 represent the year 1990.

• Write an equation that could be used to predict the cost of tuition for any given year.

• Predict what the tuition will be for the year 2015.

In this problem, we know that in the year 1991 the cost was $5000. Since these two are directly related, this is

an ordered pair. (1991, 5000). But, since x = 0 represents 1990, x = 1 represents 1991 since its one year

later, so our actual ordered pair is (1, 5000)

In the year 2012, the cost was $12,500. Again this is a direct relationship, so it’s an ordered pair.

(2012, 12500). Since = 0 represents 1990, x = 22 represents 2012 since its 22 years after 1990. (22, 12500)

Now that we have two ordered pairs: (1, 5000) & (22, 12500) we can find the slope using the slope formula.

y2- y1 = 12500 – 5000 = 7500 = 357.14

x2 – x1 22-1 21

The slope is 357.14 and 1 point is (1, 5000). Let’s us this to find the y-intercept (b).

Y = mx+ b

5000 = 357.14 (1) + b Substitute for m, x, and y.

5000 = 357.14 + b Simplify: 357.14(1) = 357.14

5000-357.14 = 357.14 – 357.14 + b Subtract 35.14 from both sides

4642.86 = b The y-intercept = 4642.86

Now use the slope and y intercept to write the equation. m = 357.14 b = 4642.86

Y = mx+ b

Y = 357.14 + 4642.86

Now using our new equation, we can predict what the tuition will be for the year 2015.

Y = 357.14x + 4642.86

Y = 357.14(25) + 4642.86 Substitute 25 for x since 2015 is 25 years beyond 1990.

Y = 13571.36

In the year 2015, the tuition for this school will be about $13571.36



15. In 2003, the cost of season football tickets was $2,200. In 2012, the cost of the same season

tickets is $4,550. Let x = 0 represent the year 2000.

• Write an equation that could be used to predict the cost of the tickets for any given year.

• Predict what the cost of the tickets will be in the year 2020.

In 2003, the cost of tickets was $2200. This is an ordered pair. Since x = 0 represents 2000, our ordered pair

is (3, 2200). 2003 is three years later than 2000.

In 2012 the tickets were $4550. This is also an ordered pair. (12, 4550) 2012 is 12 years later than 2000.

Now we will use our ordered pairs to find the slope for this equation. (3, 2200) (12, 4550)

y2- y1 = 4550 – 2200 = 2350 = 261.1 The slope is 261.1 x2 – x1 12 – 3 9 Now we will use the slope and 1 point (3, 2200) to find the y-intercept. Y = mx+b m = 261.1 x = 3 y = 2200 2200 = 261.1(3) + b Substitute for m, x, and y. 2200 = 783.3 + b Simplify: 261.1*3 = 783.3 2200 -783.3 = 783.3-783.3 +b Subtract 783.3 from both sides. 1416.70 = b The y-intercept is 1416.70

Now that we know the slope and y-intercept, we can write our equation.

Y = mx+ b

Y = 261.1x +1416.70 is the equation that can be used to predict the cost of the tickets.

Y = 261.1x + 1416.70 Equation from above

Y = 261.1(20) + 1416.70 Substitute 20 for x since 2020 is 20 years after 2000.

Y = 6638.70 Approximate cost of tickets.

The tickets will cost approximately $6638.70 in the year 2020.



Part 6: Writing Linear Equations - Point-Slope Form (Lesson 6)

16. Write an equation that has a slope of -3 and passes through the point (2,8)

17. Write an equation that passes through the points (1,9) and (3, -8). Explain the steps that you

used to solve this problem.

18. Write an equation that has a slope of 1/2 and an x-intercept of 1. Explain how you solved this

problem.

19. Write an equation that has a y-intercept of 22 and passes through the point (3, -7).

Since we are given slope and a point, we can use point-slope to write the equation.

m = -3 , x1 = 2 y1 =8

y – y1 = m(x-x1)

y – 8 = -3(x-2) Substitute the values into the equation

y – 8 = -3x +6 Distribute -3

y – 8 +8 = -3x +6 +8 Add 8 to both sides

y = -3x +14 The equation that has a slope of -3 and passes through (2,8)

�� −��

�� −��=−8 − 9

3 − 1=−17

2

Since we are given two points, we must first start by finding the slope of the line that passes through the two

points. We will use the slope formula to find the slope.

Now we can use the point-slope form to write the equation. (*You can use either point for x1 and y1)

m = -17/2, x1 = 1 y1 =9

y – y1 = m(x-x1)

y – 9 = -17/2(x-1) Substitute the values into the equation.

y-9 = -17/2x +17/2 Distribute the -17/2 throughout the parenthesis

y-9+9 = -17/2x +17/2 +9 Add 9 to both sides

y = -17/2x+ 35/2

We know m = ½ and an x-intercept of 1 means that we have the point (1,0). Use point slope form.

m = 1/2, x1 = 1 y1 =0

y – y1 = m(x-x1)

y – 0 = 1/2(x-1) Substitute the values into the equation.

y = 1/2x – 1/2

We know the y-intercept is 22, but we don’t know the slope which is essential. If we write the y intercept as a

point: (0,22) and we use the other point (3, -7) you can use the slope formula to find the slope.

��

��=

��

��=

��

� is the slope.

M = -29/3 b = 22 y = -29/3x +22 is the equation.



20. Write an equation that has an x-intercept of -5 and a y-intercept of 8.

21. Mei earned $38 on her investment in 5 months. She earned $62 on the same investment in 8

months.

• Write an equation that can be used to find the amount earned (y) on Mei’s invest in x number

of months.

• About how much will Mei earn after 15 months?

Since we are given the x and y intercept, we can write and equation in standard form pretty easily.

We know that a common multiple of 5 and 8 is 40, so that will be our constant.

Since we know (-5,0) & (0,8) are intercepts, we need to write an equation that works:

A(-5) + B(8) = 40 In order for the x and y intercepts to work, A = -8 and B = 5

-8x + 5y = 40. Or 8x – 5y = -40 if you want to make the lead coefficient work.

Justify: 8x – 5y = -40 substitute (-5,0) and (0,8) and make sure they work.

8(-5) – 5(0) = -40 8(0) – 5(8) = -40

-40 = -40 -40=-40 It works!

*You can also write an equation in slope intercept form. If you took this route, your equation would be:

Y = 8/5x +8

�� −��

�� −��=62 − 38

8 − 5= 24

3= 8

We are given two points in this problem: (5, 38) (8, 62). We can find the slope and then use point-slope

form to write the equation.

Now we can use the point-slope form to write the equation. (*You can use either point for x1 and y1)

m = 8, x1 = 5 y1 =38

y – y1 = m(x-x1)

y – 38 = 8(x-5) Substitute the slope and point coordinates into the equation.

y – 38 = 8x – 40 Distribute 8 throughout the parenthesis.

Y – 38+38 = 8x -40+38 Add 38 to both sides of the equation.

Y = 8x -2

After 15 months, Mei will have earned $118.

Y = 8x – 2

Y = 8(15) – 2

Y = 118



• About how much is Mei earning per month? Explain how you determined your answer.

Part 7: Parallel and Perpendicular Lines (Lesson 7)

22. Write an equation for a line that passes through (2,-8) and is perpendicular to a line whose slope

is 5. Explain how you determined your answer.

23. Write an equation for a line that passes through (-3,6) and is parallel to the graph of x = 5.

Parallel Lines: Have the same slope and different y-intercepts.

Perpendicular lines: The slopes are the negative reciprocal of each other.

Mei is earning about $8 per month on her investment.

Since a line that is perpendicular will have a slope that is the negative reciprocal, that means that the

slope of this line will be -1/5. Now we can use point slope form to write the equation of the line.

m = -1/5, x1 = 2 y1 =-8

y – y1 = m(x-x1)

y – -8 = -1/5(x-2) Substitute values into the equation.

y+8 = -1/5x +2/5 Distribute -1/5

y + 8 – 8 = -1/5x +2/5 – 8 Subtract 8 from both sides of the equation.

y = -1/5x – 38/5 is the line perpendicular.

X = 5 is a vertical line. Therefore, a line that is parallel is also going to be a vertical line. If it passes

through (-3,6), then it must be a vertical line through (-3,6) which means that the equation is: x = -3



24. Write an equation for a line that is perpendicular to the graph of 2x – 2y = 10 and intersects the

graph at its x-intercept.

Part 8: Scatter Plots and Line of Best Fit (Lesson 8)

25. The following data represents the total U.S. E-commerce sales from 2002-2010. (Statistic from

statista.com) Let x =0 represent the year 2000.

Year E-commerce Sales (in billions of dollars)

2002 72

2004 117

2006 171

2008 214

2010 228

• Using your graphing calculator, write an equation for the line of best fit for this data.

• What does the slope and y-intercept represent in the context of this problem.?

We must first determine the slope of 2x-2y = 10 by rewriting it in slope intercept form.

2x – 2x – 2y = -2x +10 Subtract 2x from both sides

-2y = -2x +10

-2y/-2 = -2x/-2 +10/-2 Divide all terms by -2

Y = x -5 The slope of this line is 1

Now we can use point-slope form. Slope will be -1 for the perpendicular line and passes through the x-

intercept which is 5 or (5,0). 2x = 10

X = 5

m = -1 x1 = 5 y1 =-0

y – y1 = m(x-x1)

y – 0 = -1(x-5) Substitute the given values.

Y = -x +5 Is the equation for a line perpendicular to 2x-27 = 10 and passes

through its x-intercept.

In this problem, the slope represents the amount that the E-commerce sales increase per year.

The E-commerce sales increase by 20.45 billion dollars each year.

The y-intercept represents the amount of E-commerce sales in the year 2000 because when x = 0

this represents the year 2000. Therefore, the estimated amount of E-commerce sales in 2000 was

$37.7 billion dollars.

Line of best fit is: y = 20.45x +37.7 (all numbers in billions of dollars)



• Predict the amount of E-commerce sales in the U.S. for the year 2012. Explain how you

determined your answer.

In order to predict the amount for the year 2012, we must substitute 12 into our line of best fit equation.

y = 20.45x +37.7

y = 20.45(12) +37.7

y = 283.1 billion E-commerce sales in 2012 will be about $283.1 billion dollars.

part 1: writing equations in slope intercept form. (lesson ... equations/unit 3... · 1/2x + 3y = 9...

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