A Line That Passes Through The Points { (-4, 10)$}$ And { (-1, 5)$}$ Can Be Represented By The Equation { Y = -\frac{5}{3}(x - 2)$} . W H I C H E Q U A T I O N S A L S O R E P R E S E N T T H I S L I N E ? S E L E C T T H R E E O P T I O N S . A . \[ . Which Equations Also Represent This Line? Select Three Options.A. \[ . Whi C H E Q U A T I O N S A L Sore P Rese N Tt Hi S L In E ? S E L Ec Tt H Reeo Pt I O N S . A . \[ Y =

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Introduction

In mathematics, a line can be represented in various forms, including the slope-intercept form, point-slope form, and standard form. Each form provides a unique way to express the equation of a line, and understanding these forms is essential for solving problems and visualizing the relationship between variables. In this article, we will explore the equation of a line that passes through the points (βˆ’4,10){(-4, 10)} and (βˆ’1,5){(-1, 5)} and is represented by the equation y=βˆ’53(xβˆ’2){y = -\frac{5}{3}(x - 2)}. We will then examine three alternative equations that also represent this line.

The Given Equation

The given equation of the line is y=βˆ’53(xβˆ’2){y = -\frac{5}{3}(x - 2)}. This equation is in the point-slope form, where the slope of the line is βˆ’53{-\frac{5}{3}} and the point (2,0){(2, 0)} is on the line. To understand why this equation represents the line passing through the points (βˆ’4,10){(-4, 10)} and (βˆ’1,5){(-1, 5)}, let's analyze the slope and the point.

Slope and Point

The slope of the line, βˆ’53{-\frac{5}{3}}, can be calculated using the formula m=y2βˆ’y1x2βˆ’x1{m = \frac{y_2 - y_1}{x_2 - x_1}}, where (x1,y1){(x_1, y_1)} and (x2,y2){(x_2, y_2)} are two points on the line. Using the given points (βˆ’4,10){(-4, 10)} and (βˆ’1,5){(-1, 5)}, we can calculate the slope as follows:

m=5βˆ’10βˆ’1βˆ’(βˆ’4)=βˆ’53=βˆ’53{m = \frac{5 - 10}{-1 - (-4)} = \frac{-5}{3} = -\frac{5}{3}}

This confirms that the slope of the line is indeed βˆ’53{-\frac{5}{3}}.

Point-Slope Form

The point-slope form of a line is given by the equation yβˆ’y1=m(xβˆ’x1){y - y_1 = m(x - x_1)}, where (x1,y1){(x_1, y_1)} is a point on the line and m{m} is the slope. Using the point (2,0){(2, 0)} and the slope βˆ’53{-\frac{5}{3}}, we can write the equation of the line as:

yβˆ’0=βˆ’53(xβˆ’2){y - 0 = -\frac{5}{3}(x - 2)}

Simplifying this equation, we get:

y=βˆ’53(xβˆ’2){y = -\frac{5}{3}(x - 2)}

This is the given equation of the line.

Alternative Equations

Now that we have understood the given equation, let's explore three alternative equations that also represent this line.

Equation A: Slope-Intercept Form

The slope-intercept form of a line is given by the equation y=mx+b{y = mx + b}, where m{m} is the slope and b{b} is the y-intercept. To find the y-intercept, we can substitute the point (2,0){(2, 0)} into the equation:

0=βˆ’53(2)+b{0 = -\frac{5}{3}(2) + b}

Solving for b{b}, we get:

b=103{b = \frac{10}{3}}

Therefore, the equation of the line in slope-intercept form is:

y=βˆ’53x+103{y = -\frac{5}{3}x + \frac{10}{3}}

Equation B: Standard Form

The standard form of a line is given by the equation Ax+By=C{Ax + By = C}, where A{A}, B{B}, and C{C} are constants. To find the equation of the line in standard form, we can multiply both sides of the equation by βˆ’3{-3} to eliminate the fraction:

βˆ’3y=5(xβˆ’2){-3y = 5(x - 2)}

Expanding and simplifying, we get:

βˆ’3y=5xβˆ’10{-3y = 5x - 10}

Rearranging the terms, we get:

5x+3y=10{5x + 3y = 10}

This is the equation of the line in standard form.

Equation C: Point-Slope Form with a Different Point

We can also write the equation of the line in point-slope form using a different point. Let's use the point (βˆ’4,10){(-4, 10)}. The equation of the line in point-slope form is:

yβˆ’10=βˆ’53(xβˆ’(βˆ’4)){y - 10 = -\frac{5}{3}(x - (-4))}

Simplifying this equation, we get:

yβˆ’10=βˆ’53(x+4){y - 10 = -\frac{5}{3}(x + 4)}

Expanding and simplifying, we get:

yβˆ’10=βˆ’53xβˆ’203{y - 10 = -\frac{5}{3}x - \frac{20}{3}}

Rearranging the terms, we get:

y=βˆ’53xβˆ’203+10{y = -\frac{5}{3}x - \frac{20}{3} + 10}

Simplifying further, we get:

y=βˆ’53x+103{y = -\frac{5}{3}x + \frac{10}{3}}

This is the same equation as Equation A.

Conclusion

Introduction

In our previous article, we explored the equation of a line that passes through the points (βˆ’4,10){(-4, 10)} and (βˆ’1,5){(-1, 5)} and is represented by the equation y=βˆ’53(xβˆ’2){y = -\frac{5}{3}(x - 2)}. We also examined three alternative equations that also represent this line. In this article, we will answer some of the most frequently asked questions about lines and their equations.

Q&A

Q: What is the slope of the line?

A: The slope of the line is βˆ’53{-\frac{5}{3}}. This can be calculated using the formula m=y2βˆ’y1x2βˆ’x1{m = \frac{y_2 - y_1}{x_2 - x_1}}, where (x1,y1){(x_1, y_1)} and (x2,y2){(x_2, y_2)} are two points on the line.

Q: What is the y-intercept of the line?

A: The y-intercept of the line is 103{\frac{10}{3}}. This can be found by substituting the point (2,0){(2, 0)} into the equation y=βˆ’53x+b{y = -\frac{5}{3}x + b}.

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

A: To convert the equation from point-slope form to slope-intercept form, you can use the following steps:

  1. Identify the slope and the point on the line.
  2. Substitute the point into the equation yβˆ’y1=m(xβˆ’x1){y - y_1 = m(x - x_1)}.
  3. Simplify the equation to get the slope-intercept form y=mx+b{y = mx + b}.

Q: How do I convert the equation from slope-intercept form to standard form?

A: To convert the equation from slope-intercept form to standard form, you can use the following steps:

  1. Identify the slope and the y-intercept.
  2. Substitute the slope and y-intercept into the equation Ax+By=C{Ax + By = C}.
  3. Simplify the equation to get the standard form.

Q: Can I use any point on the line to write the equation in point-slope form?

A: Yes, you can use any point on the line to write the equation in point-slope form. However, the point you choose will affect the form of the equation.

Q: How do I determine which form of the equation is most useful for a particular problem?

A: The form of the equation that is most useful for a particular problem will depend on the specific requirements of the problem. For example, if you need to find the slope of the line, the slope-intercept form may be most useful. If you need to find the equation of the line in a specific format, the standard form may be most useful.

Conclusion

In this article, we have answered some of the most frequently asked questions about lines and their equations. We hope that this information has been helpful in understanding the different forms of a line and how to use them to solve problems. If you have any further questions, please don't hesitate to ask.

Additional Resources

Practice Problems

  • Find the equation of the line that passes through the points (2,3){(2, 3)} and (4,5){(4, 5)}.
  • Convert the equation y=2xβˆ’3{y = 2x - 3} from slope-intercept form to standard form.
  • Find the slope of the line that passes through the points (βˆ’2,4){(-2, 4)} and (1,2){(1, 2)}.

We hope that these practice problems will help you to reinforce your understanding of lines and their equations.