Simplifying the Value of b

Oct 13, 2025 by miodeya.com 138 views

Introduction

In mathematics, the slope-intercept form of a line is a fundamental concept that helps us understand the relationship between the slope and the y-intercept of a line. Given a point and the slope of a line, we can use the slope-intercept form to write an equation for the line. In this article, we will explore how to write an equation in slope-intercept form for a line that passes through a given point and has a specified slope.

Understanding the Slope-Intercept Form

The slope-intercept form of a line is given by the equation:

y = mx + b

where:

m is the slope of the line
b is the y-intercept of the line
x is the independent variable
y is the dependent variable

Given Information

We are given a point (-8, 7) and the slope of the line is -5/4. We need to write an equation in slope-intercept form for this line.

Step 1: Write the Equation in Slope-Intercept Form

To write the equation in slope-intercept form, we need to substitute the given slope (m) and the point (-8, 7) into the equation.

y = mx + b

We know that the slope (m) is -5/4, so we can substitute this value into the equation:

y = (-5/4)x + b

Step 2: Find the Value of b

To find the value of b, we can substitute the point (-8, 7) into the equation:

7 = (-5/4)(-8) + b

Simplifying the Equation

To simplify the equation, we can multiply the slope (m) by the x-coordinate of the point:

7 = (5/4)(8) + b

Simplifying Further

To simplify further, we can multiply the slope (m) by the x-coordinate of the point:

7 = 10 + b

Solving for b

To solve for b, we can subtract 10 from both sides of the equation:

b = 7 - 10

Simplifying the Value of b

To simplify the value of b, we can subtract 10 from 7:

b = -3

Writing the Equation in Slope-Intercept Form

Now that we have found the value of b, we can write the equation in slope-intercept form:

y = (-5/4)x - 3

Conclusion

In this article, we have learned how to write an equation in slope-intercept form for a line that passes through a given point and has a specified slope. We have used the given slope and point to find the value of b, and then written the equation in slope-intercept form. This is a fundamental concept in mathematics that helps us understand the relationship between the slope and the y-intercept of a line.

Example Problems

Write an equation in slope-intercept form for a line that passes through the point (2, 5) and has a slope of 3/2.
Write an equation in slope-intercept form for a line that passes through the point (-3, 2) and has a slope of -2/3.

Solutions

y = (3/2)x + 1
y = (-2/3)x + 4
A Line in Slope-Intercept Form: Q&A =====================================

Introduction

In our previous article, we explored how to write an equation in slope-intercept form for a line that passes through a given point and has a specified slope. In this article, we will answer some frequently asked questions about writing equations in slope-intercept form.

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

A: The slope-intercept form of a line is given by the equation:

y = mx + b

where:

m is the slope of the line
b is the y-intercept of the line
x is the independent variable
y is the dependent variable

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

A: To find the value of b, you can substitute the point (x, y) into the equation:

y = mx + b

Then, solve for b.

Q: What if I don't have a point to substitute into the equation?

A: If you don't have a point to substitute into the equation, you can use the slope-intercept form to find the equation of the line. For example, if you know the slope (m) and the y-intercept (b), you can write the equation as:

y = mx + b

Q: Can I use the slope-intercept form to find the equation of a horizontal line?

A: Yes, you can use the slope-intercept form to find the equation of a horizontal line. A horizontal line has a slope of 0, so the equation will be:

y = b

Q: Can I use the slope-intercept form to find the equation of a vertical line?

A: No, you cannot use the slope-intercept form to find the equation of a vertical line. A vertical line has an undefined slope, so the equation will be in the form:

x = a

Q: How do I determine if a line is parallel or perpendicular to another line?

A: To determine if a line is parallel or perpendicular to another line, you can compare their slopes. If the slopes are equal, the lines are parallel. If the product of the slopes is -1, the lines are perpendicular.

Q: Can I use the slope-intercept form to find the equation of a line that passes through the origin?

A: Yes, you can use the slope-intercept form to find the equation of a line that passes through the origin. The origin is the point (0, 0), so the equation will be:

y = mx

Q: Can I use the slope-intercept form to find the equation of a line that passes through a point and has a specified slope?

A: Yes, you can use the slope-intercept form to find the equation of a line that passes through a point and has a specified slope. You can substitute the point and the slope into the equation and solve for b.

Conclusion

In this article, we have answered some frequently asked questions about writing equations in slope-intercept form. We have covered topics such as finding the value of b, using the slope-intercept form to find the equation of a horizontal or vertical line, and determining if a line is parallel or perpendicular to another line.

Example Problems

Write an equation in slope-intercept form for a line that passes through the point (2, 5) and has a slope of 3/2.
Write an equation in slope-intercept form for a line that passes through the point (-3, 2) and has a slope of -2/3.
Determine if the lines y = 2x + 1 and y = 2x - 1 are parallel or perpendicular.

Solutions

y = (3/2)x + 1
y = (-2/3)x + 4
The lines are parallel.