This Question Has Two Parts. First, Answer Part A. Then, Answer Part B.Part AWrite An Equation In Point-slope Form For The Line That Passes Through The Point \[$(-6, -3)\$\] With The Given Slope \[$m = -1\$\].

Feb 28, 2025 by ADMIN 210 views

Solving Part A: Writing an Equation in Point-Slope Form

Understanding Point-Slope Form

The point-slope form of a linear equation is given by the formula: y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope of the line. This form is useful for writing an equation of a line when we know a point on the line and the slope.

Given Information

We are given a point (-6, -3) and a slope m = -1. We need to write an equation in point-slope form for the line that passes through this point with the given slope.

Writing the Equation

To write the equation in point-slope form, we will substitute the given point (-6, -3) for (x1, y1) and the given slope m = -1 into the formula: y - y1 = m(x - x1).

Substituting the Values

Substituting (-6, -3) for (x1, y1) and m = -1, we get:

y - (-3) = -1(x - (-6))

Simplifying the Equation

Simplifying the equation, we get:

y + 3 = -1(x + 6)

Distributing the Slope

Distributing the slope -1 to the terms inside the parentheses, we get:

y + 3 = -x - 6

Rearranging the Terms

Rearranging the terms to put the equation in point-slope form, we get:

y = -x - 9

The Final Answer

The equation in point-slope form for the line that passes through the point (-6, -3) with the given slope m = -1 is:

y = -x - 9

Part B

Solving Part B: Writing an Equation in Point-Slope Form

Understanding Point-Slope Form

Given Information

We are given a point (2, 7) and a slope m = 2. We need to write an equation in point-slope form for the line that passes through this point with the given slope.

Writing the Equation

To write the equation in point-slope form, we will substitute the given point (2, 7) for (x1, y1) and the given slope m = 2 into the formula: y - y1 = m(x - x1).

Substituting the Values

Substituting (2, 7) for (x1, y1) and m = 2, we get:

y - 7 = 2(x - 2)

Simplifying the Equation

Simplifying the equation, we get:

y - 7 = 2x - 4

Rearranging the Terms

Rearranging the terms to put the equation in point-slope form, we get:

y = 2x + 3

The Final Answer

The equation in point-slope form for the line that passes through the point (2, 7) with the given slope m = 2 is:

y = 2x + 3

Conclusion

In this article, we solved two parts of a problem. In Part A, we wrote an equation in point-slope form for the line that passes through the point (-6, -3) with the given slope m = -1. In Part B, we wrote an equation in point-slope form for the line that passes through the point (2, 7) with the given slope m = 2. We used the point-slope form of a linear equation, which is given by the formula: y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope of the line.
Q&A: Point-Slope Form of a Linear Equation

Frequently Asked Questions

In this article, we will answer some frequently asked questions about the point-slope form of a linear equation.

Q1: What is the point-slope form of a linear equation?

A1: The point-slope form of a linear equation is given by the formula: y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope of the line.

Q2: How do I write an equation in point-slope form?

A2: To write an equation in point-slope form, you need to substitute the given point (x1, y1) and the given slope m into the formula: y - y1 = m(x - x1).

Q3: What is the significance of the point (x1, y1) in the point-slope form?

A3: The point (x1, y1) is a point on the line, and it is used to determine the equation of the line.

Q4: What is the significance of the slope m in the point-slope form?

A4: The slope m is the rate of change of the line, and it is used to determine the equation of the line.

Q5: Can I use the point-slope form to find the equation of a line if I know two points on the line?

A5: Yes, you can use the point-slope form to find the equation of a line if you know two points on the line. You can use the two points to find the slope of the line, and then use the point-slope form to write the equation of the line.

Q6: Can I use the point-slope form to find the equation of a line if I know the slope and one point on the line?

A6: Yes, you can use the point-slope form to find the equation of a line if you know the slope and one point on the line. You can substitute the given point and the given slope into the formula: y - y1 = m(x - x1) to write the equation of the line.

Q7: What are some common mistakes to avoid when using the point-slope form?

A7: Some common mistakes to avoid when using the point-slope form include:

Not substituting the given point and slope into the formula correctly
Not simplifying the equation correctly
Not rearranging the terms correctly to put the equation in point-slope form

Q8: Can I use the point-slope form to find the equation of a line if I know the slope and two points on the line?

A8: Yes, you can use the point-slope form to find the equation of a line if you know the slope and two points on the line. You can use the two points to find the equation of the line, and then use the point-slope form to write the equation of the line.

Conclusion

In this article, we answered some frequently asked questions about the point-slope form of a linear equation. We covered topics such as the significance of the point (x1, y1) and the slope m, how to write an equation in point-slope form, and common mistakes to avoid when using the point-slope form. We hope this article has been helpful in understanding the point-slope form of a linear equation.

Additional Resources

Practice Problems

Write an equation in point-slope form for the line that passes through the point (3, 5) with the given slope m = 2.
Write an equation in point-slope form for the line that passes through the point (1, 2) with the given slope m = -3.
Write an equation in point-slope form for the line that passes through the point (4, 6) with the given slope m = 1.

Answer Key

y - 5 = 2(x - 3)
y - 2 = -3(x - 1)
y - 6 = 1(x - 4)