Write The Slope-intercept Equation Of The Line Through The Points Created By The Ordered Pairs { (1,-3)$}$ And { (-9,77)$}$.Use { Y = Mx + B$}$ To Find The Equation.

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Introduction

In mathematics, the slope-intercept equation of a line is a fundamental concept that helps us understand the relationship between the slope and the y-intercept of a line. The slope-intercept form of a line is given by the equation y = mx + b, where m is the slope and b is the y-intercept. In this article, we will learn how to find the slope-intercept equation of a line passing through two given points.

What is the Slope-Intercept Equation?

The slope-intercept equation of a line is a mathematical representation of the line in the form y = mx + b, where:

  • m is the slope of the line, which represents the rate of change of the line.
  • b is the y-intercept of the line, which represents the point where the line intersects the y-axis.

Finding the Slope-Intercept Equation

To find the slope-intercept equation of a line passing through two given points, we can use the following steps:

Step 1: Find the Slope

The slope of a line passing through two points (x1, y1) and (x2, y2) can be found using the formula:

m = (y2 - y1) / (x2 - x1)

In our case, the two points are (1, -3) and (-9, 77). We can substitute these values into the formula to find the slope:

m = (77 - (-3)) / (-9 - 1) m = (77 + 3) / (-10) m = 80 / (-10) m = -8

Step 2: Find the Y-Intercept

Now that we have the slope, we can use one of the points to find the y-intercept. Let's use the point (1, -3). We can substitute the values of x and y into the equation y = mx + b to find the y-intercept:

-3 = (-8)(1) + b -3 = -8 + b b = 5

Step 3: Write the Slope-Intercept Equation

Now that we have the slope and the y-intercept, we can write the slope-intercept equation of the line:

y = mx + b y = (-8)x + 5

Example

Let's use the points (1, -3) and (-9, 77) to find the slope-intercept equation of the line.

m = (77 - (-3)) / (-9 - 1) m = (77 + 3) / (-10) m = 80 / (-10) m = -8

b = -3 - (-8)(1) b = -3 + 8 b = 5

y = mx + b y = (-8)x + 5

Conclusion

In this article, we learned how to find the slope-intercept equation of a line passing through two given points. We used the formula for the slope and the y-intercept to find the equation of the line. We also provided an example to illustrate the steps involved in finding the slope-intercept equation.

Tips and Tricks

  • Make sure to use the correct formula for the slope and the y-intercept.
  • Use the correct values for the points.
  • Check your calculations to ensure that you have the correct slope and y-intercept.

Frequently Asked Questions

  • What is the slope-intercept equation of a line? The slope-intercept equation of a line is a mathematical representation of the line in the form y = mx + b, where m is the slope and b is the y-intercept.
  • How do I find the slope-intercept equation of a line passing through two given points? To find the slope-intercept equation of a line passing through two given points, you can use the formula for the slope and the y-intercept.
  • What is the formula for the slope? The formula for the slope is m = (y2 - y1) / (x2 - x1).

References

Related Topics

Frequently Asked Questions

Q: What is the slope-intercept equation of a line?

A: The slope-intercept equation of a line is a mathematical representation of the line in the form y = mx + b, where m is the slope and b is the y-intercept.

Q: How do I find the slope-intercept equation of a line passing through two given points?

A: To find the slope-intercept equation of a line passing through two given points, you can use the formula for the slope and the y-intercept. The formula for the slope is m = (y2 - y1) / (x2 - x1), and the formula for the y-intercept is b = y - mx.

Q: What is the formula for the slope?

A: The formula for the slope is m = (y2 - y1) / (x2 - x1).

Q: How do I find the y-intercept?

A: To find the y-intercept, you can use the formula b = y - mx, where y is the y-coordinate of one of the points and m is the slope.

Q: What is the difference between the slope and the y-intercept?

A: The slope represents the rate of change of the line, while the y-intercept represents the point where the line intersects the y-axis.

Q: Can I use the slope-intercept equation to find the equation of a line passing through three points?

A: No, the slope-intercept equation can only be used to find the equation of a line passing through two points. If you have three points, you can use the point-slope form of a line to find the equation.

Q: How do I graph a line using the slope-intercept equation?

A: To graph a line using the slope-intercept equation, you can plot the y-intercept and then use the slope to find another point on the line.

Q: Can I use the slope-intercept equation to find the equation of a horizontal line?

A: Yes, the slope-intercept equation can be used to find the equation of a horizontal line. The slope of a horizontal line is always 0, so the equation will be in the form y = b.

Q: Can I use the slope-intercept equation to find the equation of a vertical line?

A: No, the slope-intercept equation cannot be used to find the equation of a vertical line. The slope of a vertical line is always undefined, so the equation will be in the form x = a.

Common Mistakes

  • Using the wrong formula for the slope or the y-intercept.
  • Not checking calculations to ensure that the slope and y-intercept are correct.
  • Using the slope-intercept equation to find the equation of a line passing through three points.

Tips and Tricks

  • Make sure to use the correct formula for the slope and the y-intercept.
  • Use the correct values for the points.
  • Check your calculations to ensure that you have the correct slope and y-intercept.

Real-World Applications

  • Physics: The slope-intercept equation is used to describe the motion of objects under constant acceleration.
  • Engineering: The slope-intercept equation is used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Computer Science: The slope-intercept equation is used in algorithms for computer graphics and game development.

Conclusion

The slope-intercept equation is a powerful tool for describing the relationship between the slope and the y-intercept of a line. By understanding how to use the slope-intercept equation, you can solve a wide range of problems in mathematics, physics, engineering, and computer science. Remember to use the correct formula for the slope and the y-intercept, and to check your calculations to ensure that you have the correct slope and y-intercept.