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## Digital SAT Math

### Course: Digital SAT Math > Unit 2

Lesson 4: Graphs of linear equations and functions: foundations# Graphs of linear equations and functions | Lesson

A guide to graphs of linear equations and functions on the digital SAT

## What are graphs of linear equations and functions questions?

**Graphs of linear equations and functions**questions deal with linear equations and functions and their graphs in the x, y-plane. For example, the graph of y, equals, 2, x, minus, 1 is shown below.

An equation in function notation, f, left parenthesis, x, right parenthesis, equals, 2, x, minus, 1, can also represent this line.

In this lesson, we'll learn to:

- Identify features of linear graphs from their equations
- Write linear equations based on graphical features
- Determine the equations of parallel and perpendicular lines

**You can learn anything. Let's do this!**

## What are the features of lines in the x, y-plane?

### Intro to slope

### Features of lines in the x, y-plane

#### The slope

The

**slope**of a line describes its direction and steepness.- A line that trends upward from left to right has a positive slope.
- A line that trends downward from left to right has a negative slope.
- The steeper the line is, the larger the of its slope is.

The slope is equal to the ratio of a line's change in y-value to its change in x-value. We can calculate the slope using any two points on the line, left parenthesis, x, start subscript, 1, end subscript, comma, y, start subscript, 1, end subscript, right parenthesis and left parenthesis, x, start subscript, 2, end subscript, comma, y, start subscript, 2, end subscript, right parenthesis:

**Example:**Line ell contains the points left parenthesis, minus, 1, comma, 2, right parenthesis and left parenthesis, 4, comma, 12, right parenthesis. What is the slope of line ell ?

A horizontal line has a slope of 0 since all points on the line have the same y-coordinate (so the change in y is 0).

A vertical line has an undefined slope since all points on the line have the same x-coordinate (so the change in x is 0).

#### The y-intercept

The

**y-intercept**of the line is the point where the line crosses the y-axis. This point always has an x-coordinate of 0. All non-vertical lines have exactly one y-intercept.#### The x-intercept

The

**x-intercept**of the line is the point where the line crosses the x-axis. This point always has a y-coordinate of 0. All non-horizontal lines have exactly one x-intercept.### Try it!

## How do I tell the features of lines from linear equations?

### Converting to slope-intercept form

### How do I interpret an equation in slope-intercept form?

Lines in the x, y-plane are visual representations of linear equations. The

**slope-intercept form**of a linear equation, y, equals, m, x, plus, b, tells us both the slope and the y-intercept of the line:- The slope is equal to m.
- The y-intercept is equal to b.

For example, the graph of y, equals, start color #7854ab, 3, end color #7854ab, x, start color #ca337c, minus, 7, end color #ca337c has a slope of start color #7854ab, 3, end color #7854ab and a y-intercept of start color #ca337c, minus, 7, end color #ca337c.

Because the slope-intercept form shows us the features of the line outright, it's useful to rewrite any linear equation representing a line in slope-intercept form.

**Example:**What is the slope of the graph of 3, x, plus, 4, y, equals, 12 ?

### Try it!

## How do I write linear equations based on slopes and points?

### Slope-intercept equation from two points

### What information do I need to write a linear equation?

We can write the equation of a line as long as we know either of the following:

- The slope of the line and a point on the line
- Two points on the line

In both cases, we'll be using the information provided to find the missing values in y, equals, m, x, plus, b.

#### The slope and a point

When we're given the slope and a point, we have values for x, y, and m in the equation y, equals, m, x, plus, b, and we just need to plug in the values and solve for the y-intercept b.

**Example:**If line a has a slope of 2 and passes through the point left parenthesis, 1, comma, 3, right parenthesis, what is the equation of line a ?

**Note:**if the given point

*is*the y-intercept, then we just need to plug in the slope for m and the y-intercept for b. No calculation needed!

#### Two points

When we're given two points, we must first calculate the slope using the two points, then plug in the values of x, y, and m into y, equals, m, x, plus, b to find b.

**Example:**Line b passes through the points left parenthesis, minus, 2, comma, 4, right parenthesis and left parenthesis, 1, comma, minus, 5, right parenthesis. What is the equation of line b ?

### Try it!

## How do I write equations of parallel and perpendicular lines?

### Parallel & perpendicular lines from graph

### What are the features of parallel and perpendicular lines?

In the x, y-plane, lines with different slopes will intersect exactly once.

**Parallel lines**in the x, y-plane have the same

*slope*. Parallel lines do not intersect unless they also completely overlap (i.e., are the

*same*line).

**Perpendicular lines**in the x, y-plane have slopes that are of each other. Perpendicular lines form 90, degrees angles.

The graph below shows lines ell, m, and n.

- Line start color #7854ab, ell, end color #7854ab has a slope of 2.
- Line start color #ca337c, m, end color #ca337c also has a slope of 2. It is parallel to line start color #7854ab, ell, end color #7854ab.
- Line start color #208170, n, end color #208170 has a slope of minus, start fraction, 1, divided by, 2, end fraction. It is perpendicular to both lines start color #7854ab, ell, end color #7854ab and start color #ca337c, m, end color #ca337c.

This means we can write the equation of a parallel or perpendicular line based on a slope relationship and a point on the line.

#### Let's look at some examples!

Lines p and q are graphed in the x, y-plane. Line p is represented by the equation y, equals, minus, 2, x, minus, 5. If line q is parallel to line p and passes through the point left parenthesis, 0, comma, 4, right parenthesis, what is the equation of line q ?

Line ell is represented by the equation y, equals, minus, 3, x, plus, 2. What is the equation of a line that is perpendicular to line ell and intersects line ell at left parenthesis, minus, 3, comma, 11, right parenthesis ?

### Try it!

## Your turn!

## Things to remember

The

**slope-intercept form**of a linear equation, y, equals, m, x, plus, b, tells us both the slope and the y-intercept of the line:- The slope is equal to m.
- The y-intercept is equal to b.

We can write the equation of a line as long as we know either of the following:

- The slope of the line and a point on the line
- Two points on the line

**Parallel lines**in the x, y-plane have the same slope.

**Perpendicular lines**in the x, y-plane have slopes that are negative reciprocals of each other.

## Want to join the conversation?

- By the way, when solving the first equation in the video, Sal actually misuses the formula for a slope; instead of substituting y2-y1/x2-x1, he performs y1-y2/x1-x2. Please see to it that this is resolved for it could be misleading.(7 votes)
- It doesn't matter. The two versions are interchangeable, as long as you don't confuse the order(e.g. y1-y2/x2-x1 is wrong).(19 votes)

- When two lines' slope are equal, it's said parallel. Then, Can it is said perpendicular too?(1 vote)
- No. Parallel and perpendicular are two different things. And two lines cannot be both at the same time, either.

Parallel means that they have the same slope so they never intersect.

Perpendicular means that they intersect at a right angle, meaning that their slopes are vastly different(specifically, are negative reciprocals of each other).

Therefore, two lines can never be both parallel and perpendicular at the same time.

You seem to be Korean, so I'll just add a few translations below.

Parallel -> 평행(기울기가 같고 교점이 없다.)

Perpendicular -> 수직(교점이 1개 있고 기울기가 서로 곱하면 -1이 됨, 즉 역수*-1)

결론

평행이면서 동시에 수직인 두 선은 존재할 수 없음.(20 votes)

- I certainly didn't understand a thing he said(9 votes)
- use chat gpt for the part you didn't understand.(2 votes)

- i am so confused(7 votes)
- Didn't we watch exactly the same lesson before in “Algebra”??! 😅😅(7 votes)
- why does y go down -4 when x changes by +2?

video2:36(1 vote)