Write The Point-slope Equation Of The Line With The Given Properties And Solve The Equation For Y Y Y .Given: Slope M = − 5 3 M = -\frac{5}{3} M = − 3 5 ​ , Point ( 6 , − 2 (6, -2 ( 6 , − 2 ].

Introduction

In mathematics, the point-slope equation of a line is a fundamental concept used to describe the relationship between the slope and a point on the line. Given the slope and a point on the line, we can use the point-slope equation to find the equation of the line. In this article, we will discuss how to write the point-slope equation of a line with the given properties and solve the equation for $y$.

The Point-Slope Equation

The point-slope equation of a line 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 of the line.

Given Properties

In this problem, we are given the slope $m = -\frac{5}{3}$ and a point $(6, -2)$. We can use these values to write the point-slope equation of the line.

Writing the Point-Slope Equation

To write the point-slope equation, we need to substitute the given values into the equation. We have:

$$m = -\frac{5}{3}$$

and

$$(x_1, y_1) = (6, -2)$$

Substituting these values into the point-slope equation, we get:

$$y - (-2) = -\frac{5}{3}(x - 6)$$

Simplifying the equation, we get:

$$y + 2 = -\frac{5}{3}x + 10$$

Solving the Equation for $y$

To solve the equation for $y$, we need to isolate $y$ on one side of the equation. We can do this by subtracting $2$ from both sides of the equation:

$$y = -\frac{5}{3}x + 10 - 2$$

Simplifying the equation, we get:

$$y = -\frac{5}{3}x + 8$$

Conclusion

In this article, we discussed how to write the point-slope equation of a line with the given properties and solve the equation for $y$. We used the point-slope equation formula and substituted the given values to write the equation. We then solved the equation for $y$ by isolating $y$ on one side of the equation. The final equation is:

$$y = -\frac{5}{3}x + 8$$

This equation represents the line with the given properties and can be used to find the value of $y$ for any given value of $x$.

Example Problems

Here are some example problems that you can try to practice solving the point-slope equation of a line:

• Given the slope $m = 2$ and a point $(3, 4)$, write the point-slope equation of the line and solve the equation for $y$.
• Given the slope $m = -\frac{2}{3}$ and a point $(6, -1)$, write the point-slope equation of the line and solve the equation for $y$.
• Given the slope $m = \frac{3}{4}$ and a point $(2, 3)$, write the point-slope equation of the line and solve the equation for $y$.

Tips and Tricks

Here are some tips and tricks that you can use to solve the point-slope equation of a line:

• Make sure to substitute the given values into the point-slope equation formula correctly.
• Simplify the equation by combining like terms.
• Isolate $y$ on one side of the equation by adding or subtracting the same value from both sides.
• Use the equation to find the value of $y$ for any given value of $x$.

Common Mistakes

Here are some common mistakes that you can avoid when solving the point-slope equation of a line:

• Not substituting the given values into the point-slope equation formula correctly.
• Not simplifying the equation by combining like terms.
• Not isolating $y$ on one side of the equation.
• Not using the equation to find the value of $y$ for any given value of $x$.

Conclusion

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Q: What is the point-slope equation of a line?

A: The point-slope equation of a line is a formula that describes the relationship between the slope and a point on the line. It 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 of the line.

Q: How do I write the point-slope equation of a line?

A: To write the point-slope equation of a line, you need to substitute the given values into the equation. You have:

• The slope $m$
• A point $(x_1, y_1)$ on the line

Substitute these values into the point-slope equation formula:

$$y - y_1 = m(x - x_1)$$

Q: How do I solve the point-slope equation for $y$?

A: To solve the point-slope equation for $y$, you need to isolate $y$ on one side of the equation. You can do this by adding or subtracting the same value from both sides of the equation.

For example, if you have the equation:

$$y - (-2) = -\frac{5}{3}(x - 6)$$

You can add $2$ to both sides of the equation to get:

$$y = -\frac{5}{3}x + 10$$

Q: What are some common mistakes to avoid when solving the point-slope equation?

A: Here are some common mistakes to avoid when solving the point-slope equation:

• Not substituting the given values into the point-slope equation formula correctly.
• Not simplifying the equation by combining like terms.
• Not isolating $y$ on one side of the equation.
• Not using the equation to find the value of $y$ for any given value of $x$.

Q: Can I use the point-slope equation to find the value of $x$ for any given value of $y$?

A: No, the point-slope equation is used to find the value of $y$ for any given value of $x$. If you want to find the value of $x$ for any given value of $y$, you need to use a different equation, such as the slope-intercept form of a line.

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

A: To use the point-slope equation to find the equation of a line, you need to:

1. Find the slope $m$ of the line.
2. Find a point $(x_1, y_1)$ on the line.
3. Substitute the values of $m$ and $(x_1, y_1)$ into the point-slope equation formula.
4. Simplify the equation by combining like terms.
5. Isolate $y$ on one side of the equation.

Q: Can I use the point-slope equation to find the equation of a horizontal or vertical line?

A: Yes, you can use the point-slope equation to find the equation of a horizontal or vertical line. However, you need to be careful when substituting the values of $m$ and $(x_1, y_1)$ into the equation.

For a horizontal line, the slope $m$ is $0$, and the equation is of the form:

$$y = y_1$$

For a vertical line, the slope $m$ is undefined, and the equation is of the form:

$$x = x_1$$

Q: How do I use the point-slope equation to find the equation of a line that passes through two points?

A: To use the point-slope equation to find the equation of a line that passes through two points, you need to:

1. Find the slope $m$ of the line using the two points.
2. Find one of the points $(x_1, y_1)$ on the line.
3. Substitute the values of $m$ and $(x_1, y_1)$ into the point-slope equation formula.
4. Simplify the equation by combining like terms.
5. Isolate $y$ on one side of the equation.

Conclusion

In conclusion, the point-slope equation is a powerful tool for finding the equation of a line. By using the point-slope equation formula and substituting the given values, you can write the equation of a line and solve it for $y$. With practice and patience, you can become proficient in using the point-slope equation to find the equation of a line.