Complete The Point-slope Equation Of The Line Through \[$(1,0)\$\] And \[$(6,-3)\$\]. Use Exact Numbers.$\[ Y - (-3) = \square \\]

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Introduction

In mathematics, the point-slope form of a linear equation is a powerful tool for describing the relationship between two variables. Given two points on a line, we can use the point-slope form to find the equation of the line. In this article, we will explore how to complete the point-slope equation of a line through two given points.

What is the Point-Slope Form?

The point-slope form of a linear equation 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
  • x{ x } and y{ y } are the variables

Finding the Slope

To find the slope of the line, we can use the formula:

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

where:

  • (x1,y1){ (x_1, y_1) } is the given point
  • (x2,y2){ (x_2, y_2) } is the other given point

Example

Let's say we have two points on a line: (1,0){ (1, 0) } and (6,−3){ (6, -3) }. We want to find the equation of the line through these two points.

First, we need to find the slope of the line. We can use the formula:

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

Plugging in the values, we get:

m=−3−06−1{ m = \frac{-3 - 0}{6 - 1} } m=−35{ m = \frac{-3}{5} } m=−35{ m = -\frac{3}{5} }

Completing the Point-Slope Equation

Now that we have the slope, we can complete the point-slope equation. We are given the point (1,0){ (1, 0) }, so we can plug in the values into the point-slope form:

y−0=−35(x−1){ y - 0 = -\frac{3}{5}(x - 1) }

Simplifying the equation, we get:

y=−35x+35{ y = -\frac{3}{5}x + \frac{3}{5} }

Checking the Answer

To check our answer, we can plug in the values of the two given points into the equation. Let's plug in the point (1,0){ (1, 0) }:

0=−35(1)+35{ 0 = -\frac{3}{5}(1) + \frac{3}{5} } 0=−35+35{ 0 = -\frac{3}{5} + \frac{3}{5} } 0=0{ 0 = 0 }

This shows that the equation is true for the point (1,0){ (1, 0) }.

Conclusion

In this article, we learned how to complete the point-slope equation of a line through two given points. We found the slope of the line using the formula and then plugged in the values into the point-slope form. We simplified the equation and checked our answer by plugging in the values of the two given points. With this knowledge, you can now use the point-slope form to find the equation of a line through any two given points.

Additional Examples

  • Find the equation of the line through the points (2,3){ (2, 3) } and (4,5){ (4, 5) }.
  • Find the equation of the line through the points (1,2){ (1, 2) } and (3,4){ (3, 4) }.

Tips and Tricks

  • Make sure to plug in the values of the two given points into the equation to check your answer.
  • Use the formula to find the slope of the line.
  • Simplify the equation to make it easier to read.

Common Mistakes

  • Forgetting to plug in the values of the two given points into the equation.
  • Not simplifying the equation.
  • Using the wrong formula to find the slope of the line.

Real-World Applications

  • Finding the equation of a line through two given points is a common problem in mathematics and science.
  • It can be used to model real-world situations, such as the motion of an object or the relationship between two variables.

Conclusion

Frequently Asked Questions

Q: What is the point-slope form of a linear equation? A: The point-slope form of a linear equation 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
  • x{ x } and y{ y } are the variables

Q: How do I find the slope of the line? A: To find the slope of the line, you can use the formula:

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

where:

  • (x1,y1){ (x_1, y_1) } is the given point
  • (x2,y2){ (x_2, y_2) } is the other given point

Q: What if I have two points on the line, but they are not in the correct order? A: If the points are not in the correct order, you can simply swap the x-coordinates and y-coordinates of the two points. For example, if you have the points (2,3){ (2, 3) } and (4,5){ (4, 5) }, you can swap them to get (4,5){ (4, 5) } and (2,3){ (2, 3) }.

Q: Can I use the point-slope form to find the equation of a line through three points? A: No, the point-slope form is only used to find the equation of a line through two points. If you have three points, you can use the slope formula to find the slope of the line, and then use the point-slope form to find the equation of the line.

Q: What if I have a point on the line, but I don't know the slope? A: If you have a point on the line, but you don't know the slope, you can use the point-slope form to find the equation of the line. However, you will need to know the slope of the line in order to use the point-slope form.

Q: Can I use the point-slope form to find the equation of a line through a point and a slope? A: Yes, you can use the point-slope form to find the equation of a line through a point and a slope. Simply plug in the values of the point and the slope into the point-slope form, and simplify the equation.

Q: What if I have a line with a negative slope? A: If you have a line with a negative slope, you can simply use the negative slope in the point-slope form. For example, if you have a line with a slope of -2, you can use the point-slope form:

y−y1=−2(x−x1){ y - y_1 = -2(x - x_1) }

Q: Can I use the point-slope form to find the equation of a line through a point and a vertical line? A: No, the point-slope form is only used to find the equation of a line through two points. If you have a point and a vertical line, you can use the equation of the vertical line to find the equation of the line.

Q: What if I have a line with a zero slope? A: If you have a line with a zero slope, you can simply use the point-slope form with a slope of 0. For example, if you have a line with a slope of 0, you can use the point-slope form:

y−y1=0(x−x1){ y - y_1 = 0(x - x_1) }

Q: Can I use the point-slope form to find the equation of a line through a point and a horizontal line? A: Yes, you can use the point-slope form to find the equation of a line through a point and a horizontal line. Simply plug in the values of the point and the slope into the point-slope form, and simplify the equation.

Conclusion

In conclusion, the point-slope form is a powerful tool for finding the equation of a line through two points. By following the steps outlined in this article, you can find the equation of a line through any two points. With practice and patience, you can become proficient in using the point-slope form to solve problems and model real-world situations.