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Understanding the Point-Slope Form

The point-slope form of a linear equation is a powerful tool for finding the equation of a line given two points on the line. This form is represented as y−y1=m(x−x1){y - y_1 = m(x - x_1)}, where m{m} is the slope of the line and (x1,y1){(x_1, y_1)} is a point on the line. In this article, we will use the point-slope form to find the equation of the line passing through the points (1,3){(1, 3)} and (5,1){(5, 1)}.

Finding the Slope of the Line

To find the equation of the line, we first need to find the slope of the line. The slope of a line passing through two points (x1,y1){(x_1, y_1)} and (x2,y2){(x_2, y_2)} is given by the formula:

m=y2−y1x2−x1{m = \frac{y_2 - y_1}{x_2 - x_1}}

In this case, we have the points (1,3){(1, 3)} and (5,1){(5, 1)}. Plugging these values into the formula, we get:

m=1−35−1=−24=−12{m = \frac{1 - 3}{5 - 1} = \frac{-2}{4} = -\frac{1}{2}}

Using the Point-Slope Form

Now that we have the slope of the line, we can use the point-slope form to find the equation of the line. We will use the point (1,3){(1, 3)} as the point (x1,y1){(x_1, y_1)} in the formula. Plugging in the values, we get:

y−3=−12(x−1){y - 3 = -\frac{1}{2}(x - 1)}

Simplifying the Equation

To simplify the equation, we can start by distributing the slope to the terms inside the parentheses:

y−3=−12x+12{y - 3 = -\frac{1}{2}x + \frac{1}{2}}

Next, we can add 3 to both sides of the equation to isolate the term with the variable:

y=−12x+12+3{y = -\frac{1}{2}x + \frac{1}{2} + 3}

Combining Like Terms

Now, we can combine the constant terms on the right-hand side of the equation:

y=−12x+72{y = -\frac{1}{2}x + \frac{7}{2}}

Conclusion

In this article, we used the point-slope form to find the equation of the line passing through the points (1,3){(1, 3)} and (5,1){(5, 1)}. We first found the slope of the line using the formula, and then used the point-slope form to find the equation of the line. We simplified the equation by distributing the slope and combining like terms. The final equation of the line is y=−12x+72{y = -\frac{1}{2}x + \frac{7}{2}}.

Example Problems

Problem 1

Find the equation of the line passing through the points (2,4){(2, 4)} and (6,2){(6, 2)}.

Solution

To find the equation of the line, we first need to find the slope of the line. Using the formula, we get:

m=2−46−2=−24=−12{m = \frac{2 - 4}{6 - 2} = \frac{-2}{4} = -\frac{1}{2}}

Next, we can use the point-slope form to find the equation of the line. We will use the point (2,4){(2, 4)} as the point (x1,y1){(x_1, y_1)} in the formula. Plugging in the values, we get:

y−4=−12(x−2){y - 4 = -\frac{1}{2}(x - 2)}

Simplifying the equation, we get:

y−4=−12x+1{y - 4 = -\frac{1}{2}x + 1}

Adding 4 to both sides of the equation, we get:

y=−12x+5{y = -\frac{1}{2}x + 5}

Problem 2

Find the equation of the line passing through the points (3,5){(3, 5)} and (7,3){(7, 3)}.

Solution

To find the equation of the line, we first need to find the slope of the line. Using the formula, we get:

m=3−57−3=−24=−12{m = \frac{3 - 5}{7 - 3} = \frac{-2}{4} = -\frac{1}{2}}

Next, we can use the point-slope form to find the equation of the line. We will use the point (3,5){(3, 5)} as the point (x1,y1){(x_1, y_1)} in the formula. Plugging in the values, we get:

y−5=−12(x−3){y - 5 = -\frac{1}{2}(x - 3)}

Simplifying the equation, we get:

y−5=−12x+32{y - 5 = -\frac{1}{2}x + \frac{3}{2}}

Adding 5 to both sides of the equation, we get:

y=−12x+132{y = -\frac{1}{2}x + \frac{13}{2}}

Tips and Tricks

  • When finding the slope of a line, make sure to use the correct formula.
  • When using the point-slope form, make sure to use the correct point and slope values.
  • When simplifying the equation, make sure to combine like terms and distribute the slope correctly.

Conclusion

In this article, we used the point-slope form to find the equation of the line passing through the points (1,3){(1, 3)} and (5,1){(5, 1)}. We first found the slope of the line using the formula, and then used the point-slope form to find the equation of the line. We simplified the equation by distributing the slope and combining like terms. The final equation of the line is y=−12x+72{y = -\frac{1}{2}x + \frac{7}{2}}. We also provided example problems and tips and tricks to help you master the point-slope form.

Frequently Asked Questions

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

A: The point-slope form of a linear equation is a powerful tool for finding the equation of a line given two points on the line. This form is represented as y−y1=m(x−x1){y - y_1 = m(x - x_1)}, where m{m} is the slope of the line and (x1,y1){(x_1, y_1)} is a point on the line.

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

A: To find the slope of a line using the point-slope form, 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 use the point-slope form to find the equation of a line?

A: To use the point-slope form to find the equation of a line, you need to plug in the values of the slope and the point into the formula:

y−y1=m(x−x1){y - y_1 = m(x - x_1)}

Then, simplify the equation by distributing the slope and combining like terms.

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

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

  • Not using the correct formula to find the slope
  • Not using the correct point and slope values in the formula
  • Not simplifying the equation correctly
  • Not combining like terms correctly

Q: How do I check if my answer is correct?

A: To check if your answer is correct, you can plug in the values of the point and the equation into the formula and see if it is true. You can also graph the equation and see if it passes through the point.

Q: Can I use the point-slope form to find the equation of a line with a negative slope?

A: Yes, you can use the point-slope form to find the equation of a line with a negative slope. Simply plug in the values of the slope and the point into the formula and simplify the equation.

Q: Can I use the point-slope form to find the equation of a line with a zero slope?

A: Yes, you can use the point-slope form to find the equation of a line with a zero slope. Simply plug in the values of the slope and the point into the formula and simplify the equation.

Q: Can I use the point-slope form to find the equation of a line with a fractional slope?

A: Yes, you can use the point-slope form to find the equation of a line with a fractional slope. Simply plug in the values of the slope and the point into the formula and simplify the equation.

Additional Resources

  • For more information on the point-slope form, see the article "Complete the Point-Slope Equation of the Line".
  • For more practice problems, see the article "Point-Slope Form Practice Problems".
  • For more tips and tricks, see the article "Point-Slope Form Tips and Tricks".

Conclusion

In this article, we answered some frequently asked questions about the point-slope form. We covered topics such as finding the slope of a line, using the point-slope form to find the equation of a line, and common mistakes to avoid. We also provided additional resources for further learning and practice.