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# Mastering Linear Equations: A Complete Guide with Examples and Solutions

## Introduction

In this article, we will review how to write linear equations in different forms, such as point-slope form, slope-intercept form, and standard form. We will also provide some examples and answer key for your homework practice.

## Lesson 6 Homework Practice Write Linear Equations Answer Key

### What are linear equations?

A linear equation is an algebraic equation that represents a straight line on a coordinate plane. It can have one or two variables, such as x and y. A linear equation can be written in different forms, depending on the information given or the purpose of the problem.

### Why are linear equations important?

Linear equations are important because they can model many real-world situations, such as the relationship between two quantities that change at a constant rate, the cost of a product or service based on some variables, the slope of a roof or a ramp, etc. Linear equations can also help us find the value of one variable when we know the value of another variable, or find the intersection point of two lines.

### How to write linear equations in different forms?

There are three common forms of linear equations: point-slope form, slope-intercept form, and standard form. Each form has its own advantages and disadvantages, and can be used for different purposes. We will explain each form in detail in the following sections.

## Point-slope form

### Definition and formula of point-slope form

The point-slope form of a linear equation is used when we know the slope of the line and one point on the line. The formula for point-slope form is:

y - y1 = m(x - x1)

where m is the slope of the line, and (x1, y1) is the point on the line.

### How to write an equation in point-slope form given a point and a slope

To write an equation in point-slope form given a point and a slope, we just need to plug in the values of m, x1, and y1 into the formula. For example, if we know that the slope of a line is -2, and that the line passes through the point (4, 7), then we can write the equation as:

y - 7 = -2(x - 4)

### How to write an equation in point-slope form given two points

To write an equation in point-slope form given two points, we first need to find the slope of the line using the formula:

m = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are the two points on the line. Then, we can use either point as (x1, y1) and plug in the values of m, x1, and y1 into the point-slope formula. For example, if we know that a line passes through the points (-3, 5) and (1, -1), then we can find the slope as:

m = (-1 - 5) / (1 - (-3)) = -6 / 4 = -3/2

Then, we can use either point as (x1, y1), but for simplicity, let's use (-3, 5). Then, we can write the equation as:

y - 5 = -3/2(x - (-3))

### How to convert an equation from point-slope form to slope-intercept form

The slope-intercept form of a linear equation is y = mx + b, where m is the slope of the line, and b is the y-intercept of the line. To convert an equation from point-slope form to slope-intercept form, we just need to simplify the equation by distributing the slope and adding or subtracting y1 to both sides. For example, if we have the equation:

y - 7 = -2(x - 4)

We can convert it to slope-intercept form by simplifying it as:

y - 7 = -2x + 8

y = -2x + 8 + 7

y = -2x + 15

## Slope-intercept form

### Definition and formula of slope-intercept form

The slope-intercept form of a linear equation is used when we know the slope of the line and the y-intercept of the line. The formula for slope-intercept form is:

y = mx + b

where m is the slope of the line, and b is the y-intercept of the line.

### How to write an equation in slope-intercept form given a slope and a y-intercept

To write an equation in slope-intercept form given a slope and a y-intercept, we just need to plug in the values of m and b into the formula. For example, if we know that the slope of a line is -2/3, and that the y-intercept of the line is -4, then we can write the equation as:

y = -2/3x - 4

### How to write an equation in slope-intercept form given a graph

To write an equation in slope-intercept form given a graph, we first need to identify the slope and the y-intercept of the line from the graph. The slope is the ratio of the vertical change to the horizontal change between any two points on the line. The y-intercept is the point where the line crosses the y-axis. For example, if we have this graph:

![graph](https://www.mathsisfun.com/algebra/images/slope-y-int.svg) 1/2, and that the y-intercept of the line is 3. Then, we can write the equation as:

y = -1/2x + 3

### How to write an equation in slope-intercept form given two points

To write an equation in slope-intercept form given two points, we first need to find the slope of the line using the same formula as before:

m = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are the two points on the line. Then, we can use either point as (x, y) and plug in the values of m, x, and y into the slope-intercept formula and solve for b. For example, if we know that a line passes through the points (-2, 4) and (3, -1), then we can find the slope as:

m = (-1 - 4) / (3 - (-2)) = -5 / 5 = -1

Then, we can use either point as (x, y), but for simplicity, let's use (-2, 4). Then, we can write the equation as:

y = mx + b

4 = -1(-2) + b

b = 4 - 2 = 2

y = -x + 2

## Standard form

### Definition and formula of standard form

The standard form of a linear equation is used when we want to write the equation in a simple and compact way. The formula for standard form is:

Ax + By = C

where A, B, and C are integers with no common factors, and A is non-negative.

### How to write an equation in standard form given two points

To write an equation in standard form given two points, we first need to find the slope of the line using the same formula as before:

m = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are the two points on the line. Then, we can use either point as (x, y) and plug in the values of m, x, and y into the slope-intercept formula and solve for b. Then, we can rearrange the equation to get it into standard form by multiplying both sides by a common denominator of m, moving all terms to one side, and simplifying. For example, if we know that a line passes through the points (-4, 6) and (2, -3), then we can find the slope as:

y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are the two points on the line. Then, we can use either point as (x, y) and plug in the values of m, x, and y into the slope-intercept formula and solve for b. Then, we can rearrange the equation to get it into standard form by multiplying both sides by a common denominator of m, moving all terms to one side, and simplifying. For example, if we know that a line passes through the points (-4, 6) and (2, -3), then we can find the slope as:

m = (-3 - 6) / (2 - (-4)) = -9 / 6 = -3/2

Then, we can use either point as (x, y), but for simplicity, let's use (-4, 6). Then, we can write the equation as:

y = mx + b

6 = -3/2(-4) + b

b = 6 - 6 = 0

y = -3/2x

To convert this equation to standard form, we need to multiply both sides by -2, which is the common denominator of -3/2. Then, we get:

-2y = 3x + 0

-2y - 3x = 0

This is almost in standard form, except that we need to make sure that A is non-negative. To do that, we can multiply both sides by -1. Then, we get:

2y + 3x = 0

### How to write an equation in standard form given a graph

To write an equation in standard form given a graph, we first need to identify the x-intercept and the y-intercept of the line from the graph. The x-intercept is the point where the line crosses the x-axis, and the y-intercept is the point where the line crosses the y-axis. For example, if we have this graph:

![graph](https://www.mathsisfun.com/algebra/images/standard-form.svg) We can see that the x-intercept of the line is (4, 0), and that the y-intercept of the line is (0, -2). Then, we can use these points to write an equation in slope-intercept form using the same method as before. We can find the slope as:

m = (-3 - 6) / (2 - (-4)) = -9 / 6 = -3/2

Then, we can use either point as (x, y), but for simplicity, let's use (-4, 6). Then, we can write the equation as:

y = mx + b

6 = -3/2(-4) + b

b = 6 - 6 = 0

y = -3/2x

To convert this equation to standard form, we need to multiply both sides by -2, which is the common denominator of -3/2. Then, we get:

-2y = 3x + 0

-2y - 3x = 0

This is almost in standard form, except that we need to make sure that A is non-negative. To do that, we can multiply both sides by -1. Then, we get:

2y + 3x = 0

### How to convert an equation from standard form to slope-intercept form

The slope-intercept form of a linear equation is y = mx + b, where m is the slope of the line, and b is the y-intercept of the line. To convert an equation from standard form to slope-intercept form, we just need to rearrange the equation to solve for y and then simplify. For example, if we have the equation:

2y + 3x = 0

We can convert it to slope-intercept form by solving for y as:

y = -3/2x + 0

y = -3/2x

## Conclusion

In this article, we have reviewed how to write linear equations in different forms, such as point-slope form, slope-intercept form, and standard form. We have also provided some examples and answer key for your homework practice. Here are some key takeaways from this article:

• A linear equation is an algebraic equation that represents a straight line on a coordinate plane.

• A linear equation can be written in different forms depending on the information given or the purpose of the problem.

• The point-slope form of a linear equation is used when we know the slope of the line and one point on the line.

• The slope-intercept form of a linear equation is used when we know the slope of the line and the y-intercept of the line.

• The standard form of a linear equation is used when we want to write the equation in a simple and compact way.

• We can convert an equation from one form to another by using algebraic manipulation and simplification.

## FAQs

• What is a linear equation?

• A linear equation is an algebraic equation that represents a straight line on a coordinate plane. It can have one or two variables, such as x and y.

• What are the different forms of linear equations?

• The different forms of linear equations are point-slope form, slope-intercept form, and standard form. Each form has its own advantages and disadvantages, and can be used for different purposes.

• How do you find the slope of a line?

• You can find the slope of a line by using the formula: m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on the line. The slope represents the rate of change in the y-direction as the x-coordinate changes.

• How do you write an equation in point-slope form?

• You can write an equation in point-slope form by using the formula: y - y1 = m(x - x1), where m is the slope of the line, and (x1, y1) is the point on the line.

• How do you write an equation in slope-intercept form?

• You can write an equation in slope-intercept form by using the formula: y = mx + b, where m is the slope of the line, and b is the y-intercept of the line.

• How do you write an equation in standard form?

• You can write an equation in standard form by using the formula: Ax + By = C, where A, B, and C are integers with no common factors, and A is non-negative.

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