Find The Equation Of The Line Specified.The Slope Is \[$-4\$\], And It Passes Through \[$(5,8)\$\].A. \[$y = -4x + 8\$\] B. \[$y = -4x + 28\$\] C. \[$y = -4x - 12\$\] D. \[$y = -8x + 28\$\] Please

Feb 27, 2025 by ADMIN 201 views

**Finding the Equation of a Line: A Step-by-Step Guide**

Introduction

In mathematics, the equation of a line is a fundamental concept that is used to describe the relationship between two variables. Given the slope and a point on the line, we can use this information to find the equation of the line. In this article, we will explore how to find the equation of a line using the slope-intercept form and provide a step-by-step guide on how to solve this type of problem.

What is the Slope-Intercept Form?

The slope-intercept form of a line is a mathematical equation that expresses the relationship between two variables, x and y. It is written in the form:

y = mx + b

where:

m is the slope of the line
b is the y-intercept (the point where the line intersects the y-axis)

Understanding the Problem

In this problem, we are given the slope (m = -4) and a point on the line (x = 5, y = 8). We need to find the equation of the line that passes through this point and has the given slope.

Step 1: Write the Equation in Slope-Intercept Form

The slope-intercept form of the equation is:

y = mx + b

We are given the slope (m = -4), so we can substitute this value into the equation:

y = -4x + b

Step 2: Find the Value of b

We are given a point on the line (x = 5, y = 8). We can substitute this point into the equation to find the value of b:

8 = -4(5) + b

To solve for b, we need to simplify the equation:

8 = -20 + b

Add 20 to both sides of the equation:

28 = b

Step 3: Write the Final Equation

Now that we have found the value of b, we can write the final equation of the line:

y = -4x + 28

Conclusion

In this article, we have shown how to find the equation of a line using the slope-intercept form. We have used a step-by-step guide to solve the problem and have found the equation of the line that passes through the given point and has the given slope. The final equation of the line is:

y = -4x + 28

Answer

The correct answer is:

A. y = -4x + 8

However, this is not the correct answer. The correct answer is:

B. y = -4x + 28

Why is this the correct answer?

The correct answer is B. y = -4x + 28 because we have found the value of b to be 28, and this is the value that we have substituted into the equation.

Common Mistakes

There are several common mistakes that students make when solving this type of problem. These include:

Not substituting the given point into the equation
Not simplifying the equation correctly
Not finding the value of b correctly

Tips and Tricks

Here are some tips and tricks that can help you solve this type of problem:

Make sure to substitute the given point into the equation
Simplify the equation correctly
Find the value of b correctly

Practice Problems

Here are some practice problems that you can use to practice solving this type of problem:

Find the equation of the line that passes through the point (x = 2, y = 3) and has a slope of 2.
Find the equation of the line that passes through the point (x = 4, y = 5) and has a slope of -3.
Find the equation of the line that passes through the point (x = 1, y = 2) and has a slope of 1.

Conclusion

y = -4x + 28

Q: What is the slope-intercept form of a line?

A: The slope-intercept form of a line is a mathematical equation that expresses the relationship between two variables, x and y. It is written in the form:

y = mx + b

where:

m is the slope of the line
b is the y-intercept (the point where the line intersects the y-axis)

Q: How do I find the equation of a line using the slope-intercept form?

A: To find the equation of a line using the slope-intercept form, you need to know the slope (m) and a point on the line (x, y). You can then substitute the slope and the point into the equation and solve for b.

Q: What is the y-intercept (b) in the slope-intercept form?

A: The y-intercept (b) is the point where the line intersects the y-axis. It is the value of y when x is equal to 0.

Q: How do I find the value of b in the slope-intercept form?

A: To find the value of b, you need to substitute a point on the line into the equation and solve for b. You can use the point (x, y) and the slope (m) to find the value of b.

Q: What is the slope (m) in the slope-intercept form?

A: The slope (m) is a measure of how steep the line is. It is calculated as the ratio of the change in y to the change in x.

Q: How do I calculate the slope (m) of a line?

A: To calculate the slope (m) of a line, you need to know two points on the line (x1, y1) and (x2, y2). You can then use the formula:

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

Q: What is the equation of a line that passes through the point (x = 2, y = 3) and has a slope of 2?

A: To find the equation of the line, you need to substitute the slope (m = 2) and the point (x = 2, y = 3) into the equation and solve for b.

y = 2x + b 3 = 2(2) + b 3 = 4 + b b = -1

The equation of the line is:

y = 2x - 1

Q: What is the equation of a line that passes through the point (x = 4, y = 5) and has a slope of -3?

A: To find the equation of the line, you need to substitute the slope (m = -3) and the point (x = 4, y = 5) into the equation and solve for b.

y = -3x + b 5 = -3(4) + b 5 = -12 + b b = 17

The equation of the line is:

y = -3x + 17

Q: What is the equation of a line that passes through the point (x = 1, y = 2) and has a slope of 1?

A: To find the equation of the line, you need to substitute the slope (m = 1) and the point (x = 1, y = 2) into the equation and solve for b.

y = x + b 2 = 1(1) + b 2 = 1 + b b = 1

The equation of the line is:

y = x + 1

Conclusion

In this article, we have answered some frequently asked questions about finding the equation of a line using the slope-intercept form. We have provided examples and step-by-step solutions to help you understand how to find the equation of a line. If you have any more questions or need further clarification, please don't hesitate to ask.