# Linear Relationships

## Objective

Write linear equations using slope and a given point on the line.

## Common Core Standards

### Core Standards

?

• 8.EE.B.6 — Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

• 8.F.B.4 — Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

## Criteria for Success

?

1. Use slope and a point on a line to determine where the line passes through the $y$-axis.
2. Write a linear equation in the form $y=mx+b$ using slope and $y$-intercept.
3. Write a linear equation for a word problem given rate of change and information representing an ordered pair.

## Tips for Teachers

?

Lessons 11 and 12 address writing linear equations using information about the line or situation. In Lesson 11, students are given information about the slope or rate of change, as well as information that includes a pair of $x$ and $y$ values, in order to determine the $y$-intercept or initial value.

### Remote Learning Guidance

If you need to adapt or shorten this lesson for remote learning, we suggest prioritizing Anchor Problems 2 and 3 (benefit from worked examples). Find more guidance on adapting our math curriculum for remote learning here.

#### Fishtank Plus

• Problem Set
• Student Handout Editor
• Vocabulary Package

## Anchor Problems

?

### Problem 1

Match each description of a line to the equation that represents it.

 Descriptions Equations 1. Vertical line through (4, -2) A.  $y=-2x+4$ 2. Line with slope of 2 and $y-$intercept (0, -4) B.  $y=-2+4x$ 3. Line with slope of -2 and $y-$intercept (0, 4) C.  $y=-2$ 4. Line with slope of -4 and $y-$intercept (0, 2) D.  $x=4$ 5. Line with slope of 4 and $y-$intercept (0, -2) E.  $y=-4+2x$ 6. Horizontal line through (4, -2) F.  $y=-4x+2$

### Problem 2

A line passes through the point (6, -1) and has a slope of ${-{1\over 3}}$.

What is the equation for this line in slope-intercept form?

### Problem 3

A taxicab driver charges $2.40 per mile plus a one-time flat fee. A 3-mile ride costs you$10.30.

1. Write a function to represent the cost of a taxicab ride, $y$, for $x$ miles.
2. How much will it cost you to travel 9 miles?

## Problem Set

?

The following resources include problems and activities aligned to the objective of the lesson that can be used to create your own problem set.

• Include problems where a graph is given with a non-integer y-intercept; students should still be able to find the slope from the graph.

?

### Problem 1

Throughout the summer, you save money from your summer job and put it in your savings account. Starting in the fall, you begin withdrawing $35 each week and no longer add any money to the account. After 5 weeks of withdrawing money, you have$514 left in your savings account.

How much money did you start with in your account? What function represents the amount of money in your account, $y$, after $x$ weeks of withdrawals?

Write an equation in slope-intercept form for the line that passes through the point ${(-6, -20)}$ and has a slope of $2$