WRITING LINEAR EQUATIONS

WRITING LINEAR EQUATIONS

Unit Overview

In this unit, students will be able to:

• Write linear equations from graphs, tables, and given points

• Write linear equations from graphs from tables

• Write linear equations from graphs from points on a line

• Write linear equations from word problem situation

• Write arithmetic sequences

Key Concepts

• slope-intercept form

• rate of change

• arithmetic sequence

Using the Slope-Intercept Form to Write Linear Equations

y = mx + b

m = slope of a line b = the y-intercept of a line

To write a linear equation, you need to identify two things:

1. The y-intercept (b): where the line crosses the y-axis 2. The slope (m): the steepness of the line → 𝑚 = ^rise

run

Once you identify those two items, insert them into y = mx + b

Writing a Linear Equation From a Graph

1^st: What is the y-intercept?

The line crosses the y-axis at -5 (circled in blue) 2^nd: What is the slope?

𝑚 =rise

run = up 2 right 1 = 2

1= 2

3^rd: Place the slope and y-intercept into the equation, y = mx + b y = mx + b

y = 2x + -5

y = 2x – 5 (re-write ‘adding negative 5’ as one operation: ‘minus 5’)

Let’s practice. Use the Desmos Graphing Calculator as needed.

1.) What is the slope and y-intercept of the graph below?

A. slope is -2, y-intercept is -6 B. slope is 2, y-intercept is -3 C. slope is 6, y-intercept is 2 D. slope is -3, y intercept is 6

2.) What equation represents the graph used in problem #1? (answer B)

A. y = 3x + 2 B. y = -3x + 6 C. y = 6x + 2 D. y = 2x – 3

3. Which equation represents the graph below?

A. y = 4x + 5 B. y = -4x + 5 C. 𝑦 = −5𝑥 +¹

4

D. 𝑦 = ¹

4𝑥 + 5

4. Which equation represents the graph below?

A. 𝑦 =−²₃𝑥 − 4 B. 𝑦 =²₃𝑥 − 6 C. 𝑦 =−³

2𝑥 − 4 D. 𝑦 =³₂𝑥 − 6

5. Which equation represents the graph below? (answer B)

A. y = 4 B. y = 4x C. y = x + 4 D. y = 0

Writing a Linear Equation Given Two Points

1^st: graph the two points by adding a table when using the Desmos graphing calculator (Click on the words DESMOS to open the graphing calculator link)

2^nd: identify the slope between the two points by either looking at the graph, 𝑚 = rise

run =𝑑𝑜𝑤𝑛 2 𝑢𝑛𝑖𝑡𝑠

𝑟𝑖𝑔ℎ𝑡 4 𝑢𝑛𝑖𝑡𝑠 = −2

4 = −1 2 or through algebraic calculations.

(2, -3) and (6, -5)

𝑚 = rise

run =difference between the 𝑦 − values

difference between the 𝑥 − values =−5 − (−3)

6 − 2 = −2

4 = −1 2

3^rd: identify the y-intercept by continuing the pattern of the slope as a line towards the y-axis

The extended line hits the y-axis at -2.

4^th: Place the slope and y-intercept into the equation, y = mx + b y = mx + b

𝑦 =−¹

2𝑥− 2

5^th: To check your answer, add a function on Desmos, graph 𝑦 = −¹

2𝑥 − 2, and see if it lines up with the points:

Let’s use the same example to show another method to find the equation. The Desmos

Calculator has a special function that will use the table to generate the slope and y-intercept for you.

1^st: graph the two points by adding a table when using the Desmos graphing calculator

2^nd: hit the ‘+’ icon to add a new function

3^rd: Inside that box, type a variation of the slope-intercept form: y1~ mx1+b

• the ~ symbol is to the left of the 1 key on your keyboard

• To type the smaller 1 (subscript), on your keyboard, hit ‘control’ and ‘=’ at the same time. Then type 1. After you do that, hit ‘control’ and ‘=’ at the same time to go back to normal font. Another way to do this is to simply copy and paste the x1

and y1 from the table above in Desmos.

• Once you type in y1~ mx1 + b, Desmos will give you some statistical information below, including the slope (m) and y-intercept(b). They are circled in red on the above image.

4^th: Place the slope and y-intercept into the equation, y = mx + b y = -0.5x – 2

Writing a Linear Equation Given a Table of Points

Example C: Write the linear equation represented by this table:

x y

3 -1 6 -7 7 -9 -2 9

1^st: graph the table by adding a table when using the Desmos graphing calculator

2^nd: identify the slope by choosing any two of the points. I chose (6, -7) and (7, -9).

You can then find the slope by either:

look at the graph, 𝑚 = rise

run =𝑑𝑜𝑤𝑛 2 𝑢𝑛𝑖𝑡𝑠

𝑟𝑖𝑔ℎ𝑡 1 𝑢𝑛𝑖𝑡𝑠 = −2

1 = −2 or through algebraic calculations.

(6, -7) and (7, -9)

𝑚 =rise

run = difference between the y − values

difference between the x − values= −9 − (−7)

7 − 6 =−2

1 = −2 3^rd: identify the y-intercept by continuing the pattern of the slope as a line towards the y-axis (see the green line on the above image).

The y-intercept is 5.

4^th: Place the slope and y-intercept into the equation, y = mx + b y = -2x + 5

5^th: To check your answer, add a function on Desmos, graph y = -2x + 5, and see if it lines up with the points

Let’s re-try example C using the special Desmos that gives the slope and y-intercept.

1^st: enter the table

2^nd: add a new function: type in y1~ mx1 + b

3^rd: identify the slope (m) and y-intercept given. They are circled in red on the below image.

4^th: Place the slope and y-intercept into the equation, y = mx + b y = -2x + 5

Let’s practice. Use the Desmos Graphing Calculator as needed.

6.) What is the linear equation of a line that goes through (3, -2) and (4, -5)?

A. y = 4x + 6 B. y = -3x + 7 C. y = 2x – 8 D. y = -5x + 8

7.) What is the linear equation of a line that goes through (2, 4) and (8, 7)? (y = 0.5x + 3) 8.) What is the linear equation of a line that goes through (2, 6) and (-3, -9)? (y = 3x)

9.) What is a linear equation that contains these points? (y = 2x – 7)

10.) What is a linear equation that contains these points? (𝑦 =¹

2𝑥 + 3)

11.) What is a linear equation that contains these points?

x y 3 3 5 3 8 3 -2 3 A. y = 3x B. y = 3 C. y = x + 3 D. x = 3

Write a Linear Equation From a Word Problem Scenario

o Identify your variables

o Identify your rate of change, which is your slope (m) o Identify your starting value, which is your y-intercept (b) x y

2 -3 5 3 8 9 11 15

x y -4 1 0 3 3 4.5 8 7

o Place the slope and y-intercept into the equation, y = mx + b Example D:

To join “Cory’s Health Club”, you must pay an initial $80 fee, as well as $30 per month. Write a linear equation showing the relationship between the number of months as a member (M) and the total membership cost (T)

1. Identify your variables M = # of months as member T = total membership cost

2. Identify your rate of change, which is your slope (m)

$30 per month

3. Identify your starting value, which is your y-intercept (b)

$80 initial fee

4. Place the slope and y-intercept into the equation, y = mx + b T = 30M + 80

Let’s practice.

12.) Eye-Toons charges a membership fee of $10 as well as $2.50 per cartoon downloaded.

Write a linear equation showing the relationship between the total cost (T) and the number of cartoons downloaded (C). Which equation represents the situation?

A. C = 10T + 2.50 B. T = 10C + 2.50 C. C = 2.50T + 10 D. T = 2.50C + 10

13.) Chelsey Makes $7 per hour. Write an equation showing the relationship between her total pay (P) and the number of hours she works (H). (P = 7H)

14.) Jacob has $10 to spend on pizza and drinks at the activity night. A slice of pizza (P) costs

$1.25 per slice. Each drink (D) costs $0.75. Write an equation representing the amount of drinks

& pizza he can buy?

A. P = 0.75D + 10 B. 10 = 1.25D + 0.75P C. 10 = 0.75D + 1.25P D. 2P = D + P

15.) At work, Danielle makes $9 per hour. Each paycheck, she gets $40 taken out for insurance.

Which equation represents her total pay on a paycheck (P) in relation to the hours worked (H)?

A. P = 9H – 40 B. P = 9H + 40 C. 49 = P + H D. 40 = 9H + P

16.) Which linear equation represents the relationship between the number of attendees (A) and the total cost (C)?

A. A = 4C+ 3 B. A = 3C + 4 C. C = 4A + 3 D. C = 3A + 4

Arithmetic Sequence

An arithmetic sequence is a linear pattern of numbers.

Example E:

Write a function f (x) to represent this arithmetic sequence. Then determine the 20^th term.

5, 9, 13, 17, 21,…

In order to write a function, you want to identify:

o The first term: 5

o The rate of change: 4 (4 is always added to get the next term)

5, 9, 13, 17, 21,…

Once you identify those items, you can write the function f (x) = first term + rate of change ● (x – 1)

o You multiply by x – 1 because:

▪ to get to the 2^nd term (9), you added 4 one time

▪ to get to the 3^rd term (13), you added 4 two times

▪ to get to the 4^th term (17), you added 4 three times

▪ to get to the 5^th term (21), you added 4 four times In each case, you added 4 one less time than the term # For this example, the function would be: f (x) = 5 + 4(x – 1)

You can check it by putting in 5 for x (the 5^th term). You should get the 5^th term (21).

f (5) = 5 + 4(5 – 1) = 5 + 4(4) = 5 +16 = 21 To determine the 20^th term, put 20 in for x.

f (20) = 5 + 4(20 – 1) = 81

Example F:

Write a function f(n) to represent this arithmetic sequence. Then predict the 10^th term.

28, 19, 10, 1, -8,…

The first term is 28, the rate of change is -9.

f (n) = 28 – 9(n – 1)

Check your answer for any term. The 4^th term is 1. If n = 4, the output should be 1.

f (n) = 28 – 9(4 – 1) = 28 – 9(3) = 28 – 27 = 1

To predict the 10^th term, put 10 in for n: f (10) = 28 – 9(10 – 1) = -53

Let’s practice with arithmetic sequence.

17. What is the rate of change, or common difference, for this sequence? (8) 17, 25, 33, 41, 49,…

18. Which function represents the sequence for problem #17?

A. f (x) = 8x + 17 B. f (x) = 17x + 8 C. f (x) = 17 + 8(x – 1) D. f (x) = 8 + 17(x – 1)

19. For problem #17, determine the 14^th term (121)

20. Which function represents the sequence?

40, 34, 28, 22, 16,…

A. f (x) = 40 + 6x B. f (x) = 40 – 6(x – 1) C. f (x) = 6 + 40(x – 1) D. f (x) = 6x + 40