Richard Cuts A Piece Of Wood For A Project. The First Cut Is Represented By The Equation Y = 1 2 X + 2 Y=\frac{1}{2}x+2 Y = 2 1 X + 2 . The Second Cut Needs To Be Parallel To The First And Will Pass Through The Point { (0, -7)$}$. Identify The Equation That

Mar 10, 2025 by ADMIN 260 views

**Understanding the Problem: Richard's Wood Cutting Project**

Richard is working on a project that requires him to cut a piece of wood. The first cut is represented by the equation $y=\frac{1}{2}x+2$ . This equation signifies the path that Richard has taken for his first cut. However, he needs to make a second cut that is parallel to the first one and passes through the point $(0, -7)$ . In this article, we will help Richard identify the equation that represents the second cut.

What are Parallel Lines?

Before we dive into finding the equation of the second cut, let's understand what parallel lines are. Parallel lines are lines that lie in the same plane and never intersect, no matter how far they are extended. In other words, they have the same slope but different y-intercepts.

The Equation of a Line

The equation of a line in slope-intercept form is given by $y = mx + b$ , where $m$ is the slope of the line and $b$ is the y-intercept. In the case of the first cut, the equation is $y=\frac{1}{2}x+2$ . This means that the slope of the first cut is $\frac{1}{2}$ and the y-intercept is $2$ .

Finding the Equation of the Second Cut

Since the second cut needs to be parallel to the first cut, it will have the same slope as the first cut, which is $\frac{1}{2}$ . However, it needs to pass through the point $(0, -7)$ . To find the equation of the second cut, we can use the point-slope form of a line, which is given by $y - y_1 = m(x - x_1)$ , where $(x_1, y_1)$ is a point on the line and $m$ is the slope.

Using the Point-Slope Form

Substituting the values of the point $(0, -7)$ and the slope $\frac{1}{2}$ into the point-slope form, we get:

y - (-7) = \frac{1}{2}(x - 0)

Simplifying the equation, we get:

y + 7 = \frac{1}{2}x

Subtracting $7$ from both sides, we get:

y = \frac{1}{2}x - 7

The Equation of the Second Cut

Therefore, the equation of the second cut is $y = \frac{1}{2}x - 7$ . This equation represents the path that Richard needs to take for his second cut.

Conclusion

In this article, we helped Richard identify the equation that represents the second cut of his wood cutting project. We used the concept of parallel lines and the equation of a line to find the equation of the second cut. The equation of the second cut is $y = \frac{1}{2}x - 7$ . We hope that this article has provided you with a clear understanding of how to find the equation of a line that is parallel to another line and passes through a given point.

Additional Resources

If you are interested in learning more about lines and their equations, we recommend checking out the following resources:

Frequently Asked Questions

Q: What is the equation of the first cut? A: The equation of the first cut is $y = \frac{1}{2}x + 2$ .
Q: What is the slope of the first cut? A: The slope of the first cut is $\frac{1}{2}$ .
Q: What is the y-intercept of the first cut? A: The y-intercept of the first cut is $2$ .
Q: What is the equation of the second cut? A: The equation of the second cut is $y = \frac{1}{2}x - 7$ .
Q: What is the slope of the second cut? A: The slope of the second cut is $\frac{1}{2}$ .
Q: What is the y-intercept of the second cut? A: The y-intercept of the second cut is $-7$ .
Q&A: Understanding Lines and Their Equations =============================================

In our previous article, we helped Richard identify the equation that represents the second cut of his wood cutting project. We used the concept of parallel lines and the equation of a line to find the equation of the second cut. In this article, we will answer some frequently asked questions about lines and their equations.

Q: What is the equation of a line?

A: The equation of a line in slope-intercept form is given by $y = mx + b$ , where $m$ is the slope of the line and $b$ is the y-intercept.

Q: What is the slope of a line?

A: The slope of a line is a measure of how steep it is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.

Q: How do I find the slope of a line?

A: To find the slope of a line, you can use the formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$ , where $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the line.

Q: What is the y-intercept of a line?

A: The y-intercept of a line is the point where the line intersects the y-axis. It is the value of $y$ when $x = 0$ .

Q: How do I find the y-intercept of a line?

A: To find the y-intercept of a line, you can use the equation $y = mx + b$ and substitute $x = 0$ into it. This will give you the value of $y$ when $x = 0$ .

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable is 1. For example, $y = 2x + 3$ is a linear equation. A quadratic equation, on the other hand, is an equation in which the highest power of the variable is 2. For example, $y = x^2 + 3x + 2$ is a quadratic equation.

Q: How do I graph a line?

A: To graph a line, you can use the slope-intercept form of the equation, $y = mx + b$ . You can plot the y-intercept, which is the point where the line intersects the y-axis, and then use the slope to find other points on the line.

Q: What is the equation of a line that passes through two points?

A: To find the equation of a line that passes through two points, you can use the point-slope form of the equation, $y - y_1 = m(x - x_1)$ , where $(x_1, y_1)$ and $(x_2, y_2)$ are the two points.

Q: How do I find the equation of a line that is parallel to another line?

A: To find the equation of a line that is parallel to another line, you can use the fact that parallel lines have the same slope. You can find the slope of the first line and then use it to find the equation of the second line.

Q: What is the equation of a line that is perpendicular to another line?

A: To find the equation of a line that is perpendicular to another line, you can use the fact that perpendicular lines have slopes that are negative reciprocals of each other. You can find the slope of the first line and then use it to find the slope of the second line.

Conclusion

In this article, we have answered some frequently asked questions about lines and their equations. We hope that this article has provided you with a clear understanding of the concepts and formulas involved in working with lines and their equations.

Additional Resources

If you are interested in learning more about lines and their equations, we recommend checking out the following resources:

Frequently Asked Questions

Q: What is the equation of a line? A: The equation of a line in slope-intercept form is given by $y = mx + b$ , where $m$ is the slope of the line and $b$ is the y-intercept.
Q: What is the slope of a line? A: The slope of a line is a measure of how steep it is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.
Q: How do I find the slope of a line? A: To find the slope of a line, you can use the formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$ , where $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the line.
Q: What is the y-intercept of a line? A: The y-intercept of a line is the point where the line intersects the y-axis. It is the value of $y$ when $x = 0$ .
Q: How do I find the y-intercept of a line? A: To find the y-intercept of a line, you can use the equation $y = mx + b$ and substitute $x = 0$ into it. This will give you the value of $y$ when $x = 0$ .

Glossary

Slope: A measure of how steep a line is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.
Y-intercept: The point where a line intersects the y-axis. It is the value of $y$ when $x = 0$ .
Linear equation: An equation in which the highest power of the variable is 1.
Quadratic equation: An equation in which the highest power of the variable is 2.
Point-slope form: A form of the equation of a line that uses the slope and a point on the line to find the equation.
Slope-intercept form: A form of the equation of a line that uses the slope and the y-intercept to find the equation.