Use The Point-slope Formula To Write An Equation Of The Line That Passes Through The Points { (1, -5)$}$ And { (-3, 2)$}$.Write The Answer In Slope-intercept Form (if Possible).The Equation Of The Line Is { \square$}$.

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Introduction

The point-slope formula is a powerful tool in mathematics that allows us to write an equation of a line that passes through two given points. In this article, we will explore how to use the point-slope formula to write an equation of a line that passes through the points (1,−5){(1, -5)} and (−3,2){(-3, 2)}. We will also discuss how to write the equation in slope-intercept form, if possible.

What is the Point-Slope Formula?

The point-slope formula is a mathematical formula that allows us to write an equation of a line that passes through two given points. The formula is given by:

y−y1=m(x−x1){y - y_1 = m(x - x_1)}

where (x1,y1){(x_1, y_1)} is a point on the line, m{m} is the slope of the line, and x{x} and y{y} are the coordinates of any point on the line.

Finding the Slope of the Line

To use the point-slope formula, we need to find the slope of the line that passes through the two given points. The slope of a line is given by the formula:

m=y2−y1x2−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.

In this case, we are given the points (1,−5){(1, -5)} and (−3,2){(-3, 2)}. We can use these points to find the slope of the line:

m=2−(−5)−3−1{m = \frac{2 - (-5)}{-3 - 1}} m=7−4{m = \frac{7}{-4}} m=−74{m = -\frac{7}{4}}

Using the Point-Slope Formula

Now that we have found the slope of the line, we can use the point-slope formula to write an equation of the line. We will use the point (1,−5){(1, -5)} as the point (x1,y1){(x_1, y_1)} in the formula.

y−(−5)=−74(x−1){y - (-5) = -\frac{7}{4}(x - 1)}

Simplifying the equation, we get:

y+5=−74x+74{y + 5 = -\frac{7}{4}x + \frac{7}{4}}

Writing the Equation in Slope-Intercept Form

The equation we obtained in the previous step is not in slope-intercept form. To write the equation in slope-intercept form, we need to isolate the variable y{y} on one side of the equation.

y+5=−74x+74{y + 5 = -\frac{7}{4}x + \frac{7}{4}}

Subtracting 5 from both sides of the equation, we get:

y=−74x−34{y = -\frac{7}{4}x - \frac{3}{4}}

This is the equation of the line in slope-intercept form.

Conclusion

In this article, we used the point-slope formula to write an equation of a line that passes through the points (1,−5){(1, -5)} and (−3,2){(-3, 2)}. We also discussed how to write the equation in slope-intercept form, if possible. The point-slope formula is a powerful tool in mathematics that allows us to write an equation of a line that passes through two given points. With this formula, we can solve a wide range of problems involving lines and their equations.

Example Problems

  1. Write an equation of the line that passes through the points (2,3){(2, 3)} and (−1,4){(-1, 4)}.
  2. Write an equation of the line that passes through the points (0,2){(0, 2)} and (3,−1){(3, -1)}.
  3. Write an equation of the line that passes through the points (−2,1){(-2, 1)} and (4,−3){(4, -3)}.

Solutions

  1. To write an equation of the line that passes through the points (2,3){(2, 3)} and (−1,4){(-1, 4)}, we need to find the slope of the line. The slope of the line is given by:

m=4−3−1−2{m = \frac{4 - 3}{-1 - 2}} m=1−3{m = \frac{1}{-3}} m=−13{m = -\frac{1}{3}}

Using the point-slope formula, we get:

y−3=−13(x−2){y - 3 = -\frac{1}{3}(x - 2)}

Simplifying the equation, we get:

y−3=−13x+23{y - 3 = -\frac{1}{3}x + \frac{2}{3}}

Adding 3 to both sides of the equation, we get:

y=−13x+113{y = -\frac{1}{3}x + \frac{11}{3}}

This is the equation of the line in slope-intercept form.

  1. To write an equation of the line that passes through the points (0,2){(0, 2)} and (3,−1){(3, -1)}, we need to find the slope of the line. The slope of the line is given by:

m=−1−23−0{m = \frac{-1 - 2}{3 - 0}} m=−33{m = \frac{-3}{3}} m=−1{m = -1}

Using the point-slope formula, we get:

y−2=−1(x−0){y - 2 = -1(x - 0)}

Simplifying the equation, we get:

y−2=−x{y - 2 = -x}

Adding 2 to both sides of the equation, we get:

y=−x+2{y = -x + 2}

This is the equation of the line in slope-intercept form.

  1. To write an equation of the line that passes through the points (−2,1){(-2, 1)} and (4,−3){(4, -3)}, we need to find the slope of the line. The slope of the line is given by:

m=−3−14−(−2){m = \frac{-3 - 1}{4 - (-2)}} m=−46{m = \frac{-4}{6}} m=−23{m = -\frac{2}{3}}

Using the point-slope formula, we get:

y−1=−23(x−(−2)){y - 1 = -\frac{2}{3}(x - (-2))}

Simplifying the equation, we get:

y−1=−23x−43{y - 1 = -\frac{2}{3}x - \frac{4}{3}}

Adding 1 to both sides of the equation, we get:

y=−23x−13{y = -\frac{2}{3}x - \frac{1}{3}}

Frequently Asked Questions

Q: What is the point-slope formula?

A: The point-slope formula is a mathematical formula that allows us to write an equation of a line that passes through two given points. The formula is given by:

y−y1=m(x−x1){y - y_1 = m(x - x_1)}

where (x1,y1){(x_1, y_1)} is a point on the line, m{m} is the slope of the line, and x{x} and y{y} are the coordinates of any point on the line.

Q: How do I find the slope of a line using the point-slope formula?

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

m=y2−y1x2−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: How do I write an equation of a line using the point-slope formula?

A: To write an equation of a line using the point-slope formula, you need to use the formula:

y−y1=m(x−x1){y - y_1 = m(x - x_1)}

where (x1,y1){(x_1, y_1)} is a point on the line, m{m} is the slope of the line, and x{x} and y{y} are the coordinates of any point on the line.

Q: Can I write an equation of a line in slope-intercept form using the point-slope formula?

A: Yes, you can write an equation of a line in slope-intercept form using the point-slope formula. To do this, you need to isolate the variable y{y} on one side of the equation.

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{y = mx + b}

where m{m} is the slope of the line and b{b} is the y-intercept.

Q: How do I find the y-intercept of a line?

A: To find the y-intercept of a line, you need to set x=0{x = 0} in the equation of the line and solve for y{y}.

Q: Can I use the point-slope formula to find the equation of a line that passes through three points?

A: No, you cannot use the point-slope formula to find the equation of a line that passes through three points. The point-slope formula is used to find the equation of a line that passes through two points.

Q: Can I use the point-slope formula to find the equation of a line that passes through a point and a line?

A: No, you cannot use the point-slope formula to find the equation of a line that passes through a point and a line. The point-slope formula is used to find the equation of a line that passes through two points.

Q: What are some common mistakes to avoid when using the point-slope formula?

A: Some common mistakes to avoid when using the point-slope formula include:

  • Not using the correct formula for the slope of the line
  • Not using the correct formula for the equation of the line
  • Not isolating the variable y{y} on one side of the equation
  • Not checking the work for errors

Q: How do I check my work when using the point-slope formula?

A: To check your work when using the point-slope formula, you need to:

  • Plug in the values of the points into the equation of the line
  • Simplify the equation
  • Check that the equation is in the correct form
  • Check that the equation satisfies the given conditions

Q: What are some real-world applications of the point-slope formula?

A: Some real-world applications of the point-slope formula include:

  • Finding the equation of a line that passes through two points in a coordinate plane
  • Finding the equation of a line that passes through a point and a line
  • Finding the slope of a line that passes through two points
  • Finding the equation of a line that passes through three points

Q: Can I use the point-slope formula to solve problems in other areas of mathematics?

A: Yes, you can use the point-slope formula to solve problems in other areas of mathematics, such as:

  • Algebra
  • Geometry
  • Trigonometry
  • Calculus

Q: What are some tips for using the point-slope formula effectively?

A: Some tips for using the point-slope formula effectively include:

  • Making sure to use the correct formula for the slope of the line
  • Making sure to use the correct formula for the equation of the line
  • Making sure to isolate the variable y{y} on one side of the equation
  • Making sure to check the work for errors
  • Making sure to use the point-slope formula in conjunction with other mathematical formulas and techniques.