Find The Equation In Slope-intercept Form For The Line Passing Through The Points With The Given Coordinates \[$(2,3)\$\] And \[$(-1,-1)\$\].

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Introduction

The slope-intercept form of a line is a fundamental concept in mathematics, particularly in algebra and geometry. It is a way to express the equation of a line in the form y = mx + b, where m is the slope of the line and b is the y-intercept. In this article, we will explore how to find the equation in slope-intercept form for a line passing through two given points.

What is Slope-Intercept Form?

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

  • m is the slope of the line, which is a measure of how steep the line is.
  • b is the y-intercept, which is the point at which the line intersects the y-axis.

Finding the Slope

To find the equation of a line in slope-intercept form, we need to find the slope of the line first. 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 this case, we are given two points: (2, 3) and (-1, -1). We can use these points to find the slope of the line.

Calculating the Slope

Let's calculate the slope using the formula:

m = (y2 - y1) / (x2 - x1) = (-1 - 3) / (-1 - 2) = -4 / -3 = 4/3

So, the slope of the line is 4/3.

Finding the Y-Intercept

Now that we have the slope, we can find the y-intercept by substituting one of the given points into the equation y = mx + b. Let's use the point (2, 3).

Substituting the Point

Substituting the point (2, 3) into the equation y = mx + b, we get:

3 = (4/3)(2) + b

Solving for b

Now, let's solve for b:

3 = (4/3)(2) + b 3 = 8/3 + b b = 3 - 8/3 b = (9 - 8) / 3 b = 1/3

So, the y-intercept is 1/3.

Writing the Equation in Slope-Intercept Form

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

y = mx + b y = (4/3)x + 1/3

Conclusion

In this article, we learned how to find the equation in slope-intercept form for a line passing through two given points. We used the formula for slope and substituted one of the given points into the equation to find the y-intercept. The final equation in slope-intercept form is y = (4/3)x + 1/3.

Example Use Cases

  • Finding the equation of a line passing through two points in a coordinate plane.
  • Determining the slope and y-intercept of a line.
  • Graphing a line in a coordinate plane.

Tips and Tricks

  • Make sure to use the correct formula for slope.
  • Substitute one of the given points into the equation to find the y-intercept.
  • Simplify the equation to get the final answer.

Related Topics

  • Slope of a line
  • Y-intercept of a line
  • Equation of a line in slope-intercept form

References

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Introduction

In our previous article, we learned how to find the equation in slope-intercept form for a line passing through two given points. In this article, we will answer some frequently asked questions related to this topic.

Q&A

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

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

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

A: To find the slope of a line passing through two points (x1, y1) and (x2, y2), you can use the formula:

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

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

A: The y-intercept of a line is the point at which the line intersects the y-axis. It is the value of y when x is equal to 0.

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

A: To find the y-intercept of a line passing through two points, you can substitute one of the points into the equation y = mx + b and solve for b.

Q: What is the equation of a line passing through two points?

A: The equation of a line passing through two points can be found using the slope-intercept form of a line. Once you have the slope and the y-intercept, you can write the equation of the line in the form y = mx + b.

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

A: No, the slope-intercept form 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 or the general form of a line to find the equation.

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

A: The point-slope form of a line is a way to express the equation of a line in the form y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.

Q: Can I use the point-slope form to find the equation of a line passing through two points?

A: Yes, you can use the point-slope form to find the equation of a line passing through two points. However, it is often easier to use the slope-intercept form.

Q: What is the general form of a line?

A: The general form of a line is a way to express the equation of a line in the form ax + by + c = 0, where a, b, and c are constants.

Q: Can I use the general form to find the equation of a line passing through two points?

A: Yes, you can use the general form to find the equation of a line passing through two points. However, it is often easier to use the slope-intercept form.

Conclusion

In this article, we answered some frequently asked questions related to finding the equation in slope-intercept form for a line passing through two points. We hope this article has been helpful in clarifying any confusion you may have had.

Example Use Cases

  • Finding the equation of a line passing through two points in a coordinate plane.
  • Determining the slope and y-intercept of a line.
  • Graphing a line in a coordinate plane.

Tips and Tricks

  • Make sure to use the correct formula for slope.
  • Substitute one of the given points into the equation to find the y-intercept.
  • Simplify the equation to get the final answer.

Related Topics

  • Slope of a line
  • Y-intercept of a line
  • Equation of a line in slope-intercept form
  • Point-slope form of a line
  • General form of a line

References