A Line Passes Through The Point { (-12, 3)$}$ And Has A Slope Of { -\frac{5}{6}$}$. An Equation Of The Line Is { \frac{y-3}{x+12} = -\frac{5}{6}$} . W H A T I S T H E E Q U A T I O N O F T H E L I N E I N S L O P E − I N T E R C E P T F O R M ? A . \[ .What Is The Equation Of The Line In Slope-intercept Form?A. \[ . Wha T I S T H Ee Q U A T I O N O F T H E L In E In S L O P E − In T Erce Pt F Or M ? A . \[ Y =

Introduction

In mathematics, linear equations are a fundamental concept that describe a relationship between two or more variables. The slope-intercept form of a linear equation is a powerful tool for representing and analyzing these relationships. In this article, we will explore how to convert a linear equation from point-slope form to slope-intercept form, using a specific example to illustrate the process.

Understanding Point-Slope Form

The point-slope form of a linear equation is given by the formula:

$$\frac{y-y_1}{x-x_1} = m$$

where $(x_1, y_1)$ is a point on the line, and $m$ is the slope of the line. In the given example, we have a point $(-12, 3)$ and a slope of $-\frac{5}{6}$. The equation of the line in point-slope form is:

$$\frac{y-3}{x+12} = -\frac{5}{6}$$

Converting to Slope-Intercept Form

To convert the equation from point-slope form to slope-intercept form, we need to isolate the variable $y$ on one side of the equation. The slope-intercept form of a linear equation is given by the formula:

$$y = mx + b$$

where $m$ is the slope of the line, and $b$ is the y-intercept.

To convert the equation, we can start by multiplying both sides of the equation by the denominator, $x+12$. This gives us:

$$y-3 = -\frac{5}{6}(x+12)$$

Next, we can distribute the slope to the terms inside the parentheses:

$$y-3 = -\frac{5}{6}x - 10$$

Now, we can add $3$ to both sides of the equation to isolate the variable $y$:

$$y = -\frac{5}{6}x - 10 + 3$$

Simplifying the right-hand side of the equation, we get:

$$y = -\frac{5}{6}x - 7$$

Conclusion

In this article, we have seen how to convert a linear equation from point-slope form to slope-intercept form. We used a specific example to illustrate the process, and we derived the equation of the line in slope-intercept form. The final equation is:

$$y = -\frac{5}{6}x - 7$$

This equation represents the same line as the original point-slope form, but it is now in a more convenient and useful form for analysis and application.

Key Takeaways

• The point-slope form of a linear equation is given by the formula $\frac{y-y_1}{x-x_1} = m$.
• The slope-intercept form of a linear equation is given by the formula $y = mx + b$.
• To convert an equation from point-slope form to slope-intercept form, multiply both sides of the equation by the denominator, distribute the slope, and isolate the variable $y$.

Practice Problems

1. Convert the equation $\frac{y-2}{x-3} = 2$ to slope-intercept form.
2. Convert the equation $\frac{y+1}{x+4} = -\frac{3}{4}$ to slope-intercept form.
3. Convert the equation $\frac{y-5}{x-2} = \frac{1}{2}$ to slope-intercept form.

Solutions

1. $\frac{y-2}{x-3} = 2 \Rightarrow y-2 = 2(x-3) \Rightarrow y-2 = 2x-6 \Rightarrow y = 2x-4$
2. $\frac{y+1}{x+4} = -\frac{3}{4} \Rightarrow y+1 = -\frac{3}{4}(x+4) \Rightarrow y+1 = -\frac{3}{4}x-3 \Rightarrow y = -\frac{3}{4}x-4$
3. $\frac{y-5}{x-2} = \frac{1}{2} \Rightarrow y-5 = \frac{1}{2}(x-2) \Rightarrow y-5 = \frac{1}{2}x-1 \Rightarrow y = \frac{1}{2}x+4$
   Frequently Asked Questions: Converting Linear Equations from Point-Slope Form to Slope-Intercept Form =============================================================================================

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

A: The point-slope form of a linear equation is given by the formula:

$$\frac{y-y_1}{x-x_1} = m$$

where $(x_1, y_1)$ is a point on the line, and $m$ is the slope of the line.

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

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

$$y = mx + b$$

where $m$ is the slope of the line, and $b$ is the y-intercept.

Q: How do I convert a linear equation from point-slope form to slope-intercept form?

A: To convert a linear equation from point-slope form to slope-intercept form, follow these steps:

1. Multiply both sides of the equation by the denominator.
2. Distribute the slope to the terms inside the parentheses.
3. Isolate the variable $y$ by adding or subtracting the same value from both sides of the equation.

Q: What is the y-intercept in the slope-intercept form of a linear equation?

A: The y-intercept in the slope-intercept form of a linear equation is the value of $b$ in the equation $y = mx + b$. It represents the point where the line intersects the y-axis.

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

A: To find the slope of a line given a point and a slope, use the point-slope form of a linear equation:

$$\frac{y-y_1}{x-x_1} = m$$

where $(x_1, y_1)$ is the given point, and $m$ is the given slope.

Q: Can I convert a linear equation from slope-intercept form to point-slope form?

A: Yes, you can convert a linear equation from slope-intercept form to point-slope form by rearranging the equation to isolate the variable $y$ and then rewriting it in point-slope form.

Q: What are some common mistakes to avoid when converting linear equations from point-slope form to slope-intercept form?

A: Some common mistakes to avoid when converting linear equations from point-slope form to slope-intercept form include:

• Failing to multiply both sides of the equation by the denominator.
• Failing to distribute the slope to the terms inside the parentheses.
• Failing to isolate the variable $y$ by adding or subtracting the same value from both sides of the equation.

Q: How do I check my work when converting linear equations from point-slope form to slope-intercept form?

A: To check your work when converting linear equations from point-slope form to slope-intercept form, follow these steps:

1. Plug in the values of $x$ and $y$ from the original equation into the new equation.
2. Simplify the new equation to ensure that it is equivalent to the original equation.
3. Check that the new equation has the same slope and y-intercept as the original equation.

Q: Can I use a calculator to convert linear equations from point-slope form to slope-intercept form?

A: Yes, you can use a calculator to convert linear equations from point-slope form to slope-intercept form. However, be sure to check your work by plugging in the values of $x$ and $y$ from the original equation into the new equation and simplifying the new equation to ensure that it is equivalent to the original equation.