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## SAT

### Course: SAT > Unit 6

Lesson 3: Heart of Algebra: lessons by skill- Solving linear equations and linear inequalities | Lesson
- Understanding linear relationships | Lesson
- Linear inequality word problems | Lesson
- Graphing linear equations | Lesson
- Systems of linear inequalities word problems | Lesson
- Solving systems of linear equations | Lesson
- Systems of linear equations word problems | Lesson

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# Graphing linear equations | Lesson

## What are "Graphing linear equations" questions, and how frequently do they appear on the test?

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

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
- Identify solutions to systems of linear inequalities as regions in the x, y-plane

On your official SAT, you'll likely see

**3 to 4 questions**that test your ability to graph linear equations.**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!

## How do I graph systems of linear inequalities?

### Intro to graphing systems of inequalities

### How do I identify the region representing a system of linear inequalities?

Two intersecting lines divide the x, y-plane into four regions. Points in each of the four regions represent solutions to a different system of linear inequalities.

In the graph below, the equations of the lines defining the four regions are:

Replacing the equal signs with inequality signs lets us specify one of the four regions.

When we have linear inequalities in slope-intercept form:

- If y is greater than the m, x, plus, b, shade above the line.
- If y is less than m, x, plus, b, shade below the line.

For example, let's look at the following system:

This means the solutions to the system of linear inequalities are represented by the region

*below*the line y, equals, 3, x, minus, 2 and*above*the line y, equals, start fraction, 1, divided by, 3, end fraction, x, plus, start fraction, 4, divided by, 3, end fraction:### 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.

When we have linear inequalities in slope-intercept form:

- If y is greater than the m, x, plus, b, shade above the line.
- If y is less than m, x, plus, b, shade below the line.

## Want to join the conversation?

- Please correct me if I'm missing something. But in the last example above, shouldn't the slope-intercept equation for the first line be y is less than or equal to x minus 4, rather than y is less than or equal to x plus 4? Both the computation -- multiplying 4 by -1, and the graph itself, in which the y intercept is shown to be -4, seem to suggest that. Thanks!(56 votes)
- You're absolutely right, good eye! To let the people who can fix this know, you can report the mistake in the article by going into the "Ask a question..." box and clicking "Report a Mistake" near the bottom.(22 votes)

- It appears that Sal from time to time does not adhere to the slope equation "y2-y1 over x2-x1". Rather he may do y1-y2 over x1-x2. Did he make a mistake or have I missed something?(10 votes)
- That's not a mistake in the math or anything. What you call x1 and x2 doesn't matter, as long as you subtract them in the same order on both sides of the division sign. Maybe Sal didn't feel up to remembering the name convention that we used and labeled them wrong. It doesn't change the math or the process, though.(14 votes)

- For the last equation, “ PRACTICE: MATCH A SYSTEM OF LINEAR INEQUALITIES TO ITS GRAPH” , I don’t get why 4 multiplied by -1 is still 4?(2 votes)
- When you convert the inequality "x - y >= 4" to y = mx + b form by subtracting x and then dividing by -1, you do end up with a -4. This is why when we look at the graphs, we see that the y-intercept is negative 4, not positive 4. You're right, 4 multiplied by -1 is -4 just as it should be.(2 votes)

- In the explanation for the last exercise on systems of linear inequalities, 4 multiplied times -1 should become -4.(1 vote)
- You are correct, it should be -4.(2 votes)

- From what I've seen in the fourth video the equation order doesn't matter or is it a mistake made by Sal?(1 vote)
- If you're talking about the order of equations within a system of equations, then you're right. It doesn't matter which equation is first or which one is second because you treat them both the same and can switch around their positions without anything changing.(2 votes)

- how to create a table with the linear equation(1 vote)
- When you say "create a table," do you mean an x,y table of data, as in a list of the x values and their corresponding y values?

You can create an x,y table of values by:

1) Creating your linear equation

2) Choosing your x values

3) Solving the equation for y

Repeat with as many values of x as you like, and create your table!

Hope that helped, and feel free to ask if there was something else you wanted to know!(1 vote)

- What is difference between SAT and LCAT (LUMS Common Admission Test)? I mean,to what extent do they differ?(0 votes)