A Line Passes Through The Point ( − 6 , − 4 (-6, -4 ( − 6 , − 4 ] And Has A Slope Of 5 2 \frac{5}{2} 2 5 .Write An Equation In Slope-intercept Form For This Line.

Mar 11, 2025 by ADMIN 165 views

A Line with a Given Point and Slope: Writing an Equation in Slope-Intercept Form

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Introduction

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 known slope. The slope-intercept form of a line is given by the equation y = mx + b, where m is the slope of the line and b is the y-intercept. We will use the given point and slope to find the equation of the line.

The Given Point and Slope

The given point is (-6, -4) and the slope of the line is $\frac{5}{2}$ . We can use this information to write the equation of the line in slope-intercept form.

Writing the Equation in Slope-Intercept Form

To write the equation in slope-intercept form, we need to use the point-slope form of a line, which is given by the equation y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope. We can then simplify the equation to get it in slope-intercept form.

Step 1: Plug in the Given Point and Slope

We will plug in the given point (-6, -4) and the slope $\frac{5}{2}$ into the point-slope form of a line.

y - (-4) = $\frac{5}{2}$ (x - (-6))

Step 2: Simplify the Equation

We will simplify the equation by distributing the slope and combining like terms.

y + 4 = $\frac{5}{2}$ (x + 6)

y + 4 = $\frac{5}{2}$ x + 15

Step 3: Write the Equation in Slope-Intercept Form

We will write the equation in slope-intercept form by isolating y.

y = $\frac{5}{2}$ x + 15 - 4

y = $\frac{5}{2}$ x + 11

Conclusion

We have written the equation of the line in slope-intercept form using the given point and slope. The equation is y = $\frac{5}{2}$ x + 11.

Example

Let's use the equation to find the y-coordinate of the point where x = 2.

y = $\frac{5}{2}$ (2) + 11

y = 5 + 11

y = 16

Therefore, the y-coordinate of the point where x = 2 is 16.

Applications

The equation of a line in slope-intercept form has many applications in mathematics and real-world problems. Some examples include:

Finding the equation of a line that passes through two points
Finding the equation of a line that is perpendicular to another line
Finding the equation of a line that is parallel to another line
Solving systems of linear equations

Tips and Tricks

Here are some tips and tricks to help you write the equation of a line in slope-intercept form:

Make sure to use the correct slope and point
Use the point-slope form of a line to write the equation
Simplify the equation by distributing the slope and combining like terms
Write the equation in slope-intercept form by isolating y

Common Mistakes

Here are some common mistakes to avoid when writing the equation of a line in slope-intercept form:

Using the wrong slope or point
Not simplifying the equation
Not writing the equation in slope-intercept form
Making errors when distributing the slope or combining like terms

Conclusion

Writing the equation of a line in slope-intercept form is an important skill in mathematics. By following the steps outlined in this article, you can write the equation of a line that passes through a given point and has a known slope. Remember to use the point-slope form of a line, simplify the equation, and write the equation in slope-intercept form by isolating y. With practice and patience, you can become proficient in writing the equation of a line in slope-intercept form.

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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 known slope. In this article, we will answer some frequently asked questions about writing the equation of a line in slope-intercept form.

Q&A

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 and b is the y-intercept.

Q: How do I find 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 a line, which is given by the equation y - y1 = m(x - x1), where (x1, y1) is one of the points and m is the slope. You can then simplify the equation to get it in slope-intercept form.

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

A: The point-slope form of a line is given by the equation y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.

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

A: To find the slope of a line, you can use the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) 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 given by the value of b in the equation y = mx + b.

Q: How do I write the equation of a line in slope-intercept form?

A: To write the equation of a line in slope-intercept form, you can use the point-slope form of a line and simplify the equation to get it in slope-intercept form.

Q: What are some common mistakes to avoid when writing the equation of a line in slope-intercept form?

A: Some common mistakes to avoid when writing the equation of a line in slope-intercept form include using the wrong slope or point, not simplifying the equation, and not writing the equation in slope-intercept form.

Q: How do I find 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 the slopes of perpendicular lines are negative reciprocals of each other.

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 the slopes of parallel lines are equal.

Example Problems

Problem 1: Find the equation of a line that passes through the points (2, 3) and (4, 5).

y - 3 = (5 - 3) / (4 - 2)(x - 2)

y - 3 = 2 / 2(x - 2)

y - 3 = x - 2

y = x + 1

Problem 2: Find the equation of a line that has a slope of 2 and passes through the point (3, 4).

y - 4 = 2(x - 3)

y - 4 = 2x - 6

y = 2x - 2

Problem 3: Find the equation of a line that is perpendicular to the line y = 2x + 3.

The slope of the line y = 2x + 3 is 2. The slope of a line perpendicular to this line is the negative reciprocal of 2, which is -1/2.

y - 3 = (-1/2)(x - 0)

y - 3 = -1/2x

y = -1/2x + 3

Conclusion

Writing the equation of a line in slope-intercept form is an important skill in mathematics. By following the steps outlined in this article and practicing with example problems, you can become proficient in writing the equation of a line in slope-intercept form. Remember to use the point-slope form of a line, simplify the equation, and write the equation in slope-intercept form by isolating y. With practice and patience, you can become a master of writing the equation of a line in slope-intercept form.