## Key Concepts

- Graph a linear equation
- Write an equation from a graph
- Understand slope intercept form
- Interpret slope and y- Intercept

## Slope – Intercept Form

### Slope

The slope of a line is the ratio of the amount that *y* increases as *x* increases some amount. Slope tells you how steep a line is or how much *y* increases as* x* increases.

The slope is constant (the same) anywhere on the line.

#### Slope formula

m = rise / run = y_{2}−y_{1} / x_{2}−x_{1}

#### Real-life example of slope

#### Slope-Intercept Form

The meaning of slope-intercept form is the equation of a straight line in the form y = mx + b, where *m* is the slope of the line and *b* is its y-intercept.

y = mx + b

### Graph a linear equation

#### Linear equation

A linear equation is an algebraic equation of the form y=mx+b.

Involving only a constant and a first-order (linear) term, where *m* is the slope and *b* is the y-intercept. Occasionally, the above is called a “linear equation of two variables,” where *y* and *x *are the variables.

**Example 1:**

What is the graph of y = 3x+1?

**Sol:**

**Step 1:**

Identify the y-intercept in the equation.

The y-intercept is 1, So plot the points (0, 1).

**Step 2:**

Use the slope to plot a second point.

m = 3 = vertical change / horizontal change

Start at (0, 1), move 3 units up and 1 unit to the right to locate a second point.

Plot the points (1, 4).

**Step 3:**

Draw a line through the points.

### Write an equation from a graph

**Example:**

What is the equation of the line in slope-intercept form?

**Sol:**

**Step 1:**

Find the slope between two points on the line.

The line passes through (-3, 3) and (3, -1).

Slope =

y_{2}−y_{1 }/ x_{2}−x_{1}

= (−1)−3 / 3− (−3)

=−4 / 6

= –2 / 3

**Step 2:**

Find the y–intercept.

The line intersects the y-axis at (-3, 3) so the y-intercept is 1.

**Step 3:**

Write the equation in the form *y=mx+b.*

Substitute −2 / 3 for *m* and 1 for *b.*

The equation of the line in the slope-intercept form is,

y = 2 / 3 x + 1

### Understand Slope–Intercept form

**Example:**

Write the equation of the line that passes through the given points (2, 2) and (3, 4).

**Sol:**

**Step 1:**

Find the slope of the line.

m = y_{2}−y_{1} / x_{2}−x_{1}

m **=** 4 − 2 / 3 − 2

m = 2

**Step 2:**

Use the slope and one point to find the y-intercept.

y = mx +b

4 = 2(3) +b Substitute 2 for *m*. (3, 4) for (x, y)

4 = 6 +b Simplify

-2 = b

**Step 3:**

Use the slope and y-Intercept to write the equation.

y = mx +b

y = 2x -2 Substitute 2 for *m* and -2 for *b*.

The equation in slope–intercept form of the line that passes through (2, 2) and (3, 4) is y = 2x -2.

### Interpret Slope and y-Intercept

**Example:**

The rent charged for space in an office building as a linear relationship related to the size of the space rented. Write an equation in slope-intercept form for rent at west main street office rentals

**Sol:**

**Step 1:**

Create a table or identify ordered pairs from the problem.

(600, 750) and (900, 1150)

**Step 2:**

Find the slope using the slope formula,

m = y_{2}−y_{1} / x_{2}−x_{1}

m= 1150−750 / 900−600

m= 400 / 300

m= 4 / 3

**Step 3:**

Write the form y= mx+b.

**Step 4:**

Replace *m* with the value you found.

Y= 4 / 3 x+b

**Step 5:**

Plug-in one of the points you already know and solve for *b* using the inverse operation.

750 = 4 / 3 (600)+b

750 = 800 +b

-50=b

**Step 6:**

Replace *m* and *b* with the values.

y= 4 / 3 x+(-50)

Y= 4 / 3 x -50.

## Exercise

- Sketch the graph of the equation.

y = 2x -5

- Identify the slope and y-intercept of the line for the equation.

Y = – 5x – 3/4

- Write the equation of the line that passes through the given points (0, 1) and (2, 2).
- Write the equation of each line in slope–intercept form.

- Jordan will hike the trail shown at a rate of 4 mi/h. Write a linear equation to represent the distance Jordan still has to walk after
*x*hours. What does the y-intercept of the equation represent?

- Write the equation of the line that passes through the given points (5, 4) and (-1, 6).
- Write the equation of the line in slope–intercept form.

- Graph the following line:
- Y = x + 7.

- Find the equation of the line graphed below.

- Shriya read a book cover to cover in a single session, at a constant rate. After reading for 1.5 hours, she had 402 out of the total 480 pages left to read. Let
*y*represent the number of pages left to read after*x*hours. Complete the equation for the relationship between the number of pages left and the number of hours. y=___________.

### Concept Map

### What have we learned

- Understand Slope –Intercept Form
- Graph a linear equation
- Understand how to write an equation from a graph
- Interpret slope and y- Intercept

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