A Line Passing Through The Point { (12, -5)$}$ Has A Slope Of { \frac 1}{3}$}$.Complete The Work Shown 1. Substitute Known Values For { M$ $, { X_1$}$, And { Y_1$} : \[ : \[ : \[ Y - (-5) = \frac{1}{3}(x -

Introduction

In mathematics, a line is defined by its slope and a point it passes through. The slope of a line is a measure of how steep it is, and it can be calculated using the coordinates of two points on the line. In this article, we will explore how to find the equation of a line that passes through a given point with a known slope.

The Equation of a Line

The equation of a line can be written in the form:

$$y = mx + b$$

where $m$ is the slope of the line, $x$ is the x-coordinate of a point on the line, and $b$ is the y-intercept of the line.

Substituting Known Values

We are given that the line passes through the point $(12, -5)$ and has a slope of $\frac{1}{3}$. We can substitute these values into the equation of the line:

$$y - (-5) = \frac{1}{3}(x - 12)$$

Simplifying the Equation

To simplify the equation, we can start by evaluating the expression on the left-hand side:

$$y - (-5) = y + 5$$

So, the equation becomes:

$$y + 5 = \frac{1}{3}(x - 12)$$

Isolating y

To isolate $y$, we can subtract 5 from both sides of the equation:

$$y = \frac{1}{3}(x - 12) - 5$$

Simplifying Further

We can simplify the equation further by distributing the $\frac{1}{3}$ to the terms inside the parentheses:

$$y = \frac{1}{3}x - 4 - 5$$

$$y = \frac{1}{3}x - 9$$

Conclusion

In this article, we have shown how to find the equation of a line that passes through a given point with a known slope. We started with the equation of a line in the form $y = mx + b$ and substituted the known values of the slope and the point into the equation. We then simplified the equation to isolate $y$ and arrived at the final equation of the line.

Example Use Case

Suppose we want to find the equation of a line that passes through the point $(8, 2)$ and has a slope of $2$. We can use the same steps as before to find the equation of the line:

$$y - 2 = 2(x - 8)$$

$$y - 2 = 2x - 16$$

$$y = 2x - 14$$

Tips and Tricks

• When substituting known values into the equation of a line, make sure to use the correct values for the slope and the point.
• When simplifying the equation, make sure to distribute the coefficients to the terms inside the parentheses.
• When isolating $y$, make sure to subtract the constant term from both sides of the equation.

Common Mistakes

• Failing to substitute the correct values for the slope and the point into the equation of a line.
• Failing to simplify the equation properly.
• Failing to isolate $y$ correctly.

Conclusion

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Introduction

In our previous article, we explored how to find the equation of a line that passes through a given point with a known slope. In this article, we will answer some common questions related to this topic.

Q: What is the slope of a line?

A: The slope of a line is a measure of how steep it is. It can be calculated using the coordinates of two points on the line. The slope is usually denoted by the letter $m$.

Q: How do I find the equation of a line that passes through a given point with a known slope?

A: To find the equation of a line that passes through a given point with a known slope, you need to substitute the known values into the equation of a line and simplify the equation to isolate $y$. The equation of a line is given by:

$$y = mx + b$$

where $m$ is the slope of the line, $x$ is the x-coordinate of a point on the line, and $b$ is the y-intercept of 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 usually denoted by the letter $b$.

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

A: To find the y-intercept of a line, you need to substitute the known values into the equation of a line and simplify the equation to isolate $b$. The equation of a line is given by:

$$y = mx + b$$

where $m$ is the slope of the line, $x$ is the x-coordinate of a point on the line, and $b$ is the y-intercept of the line.

Q: What is the difference between the slope and the y-intercept of a line?

A: The slope of a line is a measure of how steep it is, while the y-intercept of a line is the point where the line intersects the y-axis. The slope is usually denoted by the letter $m$, while the y-intercept is usually denoted by the letter $b$.

Q: How do I use the equation of a line to find the x-coordinate of a point on the line?

A: To find the x-coordinate of a point on the line, you need to substitute the known values into the equation of a line and simplify the equation to isolate $x$. The equation of a line is given by:

$$y = mx + b$$

where $m$ is the slope of the line, $x$ is the x-coordinate of a point on the line, and $b$ is the y-intercept of the line.

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

A: To find the equation of a line that passes through the point $(2, 3)$ and has a slope of $2$, you need to substitute the known values into the equation of a line and simplify the equation to isolate $y$. The equation of a line is given by:

$$y = mx + b$$

where $m$ is the slope of the line, $x$ is the x-coordinate of a point on the line, and $b$ is the y-intercept of the line.

Substituting the known values, we get:

$$3 = 2(2) + b$$

Simplifying the equation, we get:

$$3 = 4 + b$$

Subtracting 4 from both sides, we get:

$$-1 = b$$

So, the equation of the line is:

$$y = 2x - 1$$

Conclusion

In conclusion, finding the equation of a line that passes through a given point with a known slope is a straightforward process that involves substituting the known values into the equation of a line and simplifying the equation to isolate $y$. By following the steps outlined in this article, you can find the equation of a line that passes through a given point with a known slope.

Tips and Tricks

• When substituting known values into the equation of a line, make sure to use the correct values for the slope and the point.
• When simplifying the equation, make sure to distribute the coefficients to the terms inside the parentheses.
• When isolating $y$, make sure to subtract the constant term from both sides of the equation.

Common Mistakes

• Failing to substitute the correct values for the slope and the point into the equation of a line.
• Failing to simplify the equation properly.
• Failing to isolate $y$ correctly.

Conclusion

In conclusion, finding the equation of a line that passes through a given point with a known slope is a straightforward process that involves substituting the known values into the equation of a line and simplifying the equation to isolate $y$. By following the steps outlined in this article, you can find the equation of a line that passes through a given point with a known slope.