Given The Set Of Information, Find A Linear Equation Satisfying The Conditions, If Possible. (If Not Possible, Enter IMPOSSIBLE.)The Line Passes Through { (x, Y) = (1, 5)$}$ And { (x, Y) = (4, 14)$} . . . { Y =\$}

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Introduction

In mathematics, a linear equation is a fundamental concept that represents a straight line on a coordinate plane. Given two points on the line, we can find the equation of the line using the slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept. In this article, we will discuss how to find a linear equation satisfying the conditions given two points on the line.

The Problem

We are given two points on the line: (1, 5) and (4, 14). Our goal is to find a linear equation that passes through these two points. If it is not possible to find a linear equation, we will enter "IMPOSSIBLE" as the solution.

Finding the Slope

To find the equation of the line, we need to find the slope (m) first. The slope can be calculated using the formula:

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

where (x1, y1) and (x2, y2) are the two given points.

In this case, we have (x1, y1) = (1, 5) and (x2, y2) = (4, 14). Plugging these values into the formula, we get:

m = (14 - 5) / (4 - 1) m = 9 / 3 m = 3

Finding the Y-Intercept

Now that we have the slope (m = 3), we can find the y-intercept (b) using one of the given points. Let's use the point (1, 5).

We know that the equation of the line is y = mx + b. Plugging in the values of m and the point (1, 5), we get:

5 = 3(1) + b 5 = 3 + b b = 2

Writing the Linear Equation

Now that we have the slope (m = 3) and the y-intercept (b = 2), we can write the linear equation in slope-intercept form:

y = 3x + 2

Conclusion

In this article, we discussed how to find a linear equation satisfying the conditions given two points on the line. We used the slope-intercept form of a linear equation (y = mx + b) and found the slope (m) and y-intercept (b) using the given points. We then wrote the linear equation in slope-intercept form. The final answer is the linear equation y = 3x + 2.

Example Use Case

Suppose we want to find the equation of a line that passes through the points (2, 7) and (5, 17). We can use the same method as before to find the slope and y-intercept. The slope is:

m = (17 - 7) / (5 - 2) m = 10 / 3 m = 3.33

Using the point (2, 7), we can find the y-intercept:

7 = 3.33(2) + b 7 = 6.66 + b b = 0.34

The linear equation is:

y = 3.33x + 0.34

Tips and Tricks

  • When finding the slope, make sure to use the correct formula: m = (y2 - y1) / (x2 - x1).
  • When finding the y-intercept, make sure to use the correct point and the equation y = mx + b.
  • If the slope is undefined (i.e., the denominator is zero), it means that the line is vertical and cannot be written in slope-intercept form.

Common Mistakes

  • Failing to use the correct formula for the slope.
  • Failing to use the correct point when finding the y-intercept.
  • Not checking if the line is vertical before writing the equation in slope-intercept form.

Conclusion

In conclusion, finding a linear equation satisfying the conditions given two points on the line is a straightforward process that involves finding the slope and y-intercept using the given points. We can then write the linear equation in slope-intercept form. By following the steps outlined in this article, we can find the equation of a line that passes through any two given points.

Introduction

In our previous article, we discussed how to find a linear equation satisfying the conditions given two points on the line. In this article, we will answer some frequently asked questions related to this topic.

Q: What if the two points are the same?

A: If the two points are the same, it means that the line is a single point and cannot be written in slope-intercept form. In this case, we cannot find a linear equation that satisfies the conditions.

Q: What if the two points are on a vertical line?

A: If the two points are on a vertical line, it means that the line is vertical and cannot be written in slope-intercept form. In this case, we cannot find a linear equation that satisfies the conditions.

Q: How do I know if the line is vertical or not?

A: To determine if the line is vertical or not, you can check if the x-coordinates of the two points are the same. If they are the same, the line is vertical.

Q: Can I use any two points to find the linear equation?

A: Yes, you can use any two points to find the linear equation. However, make sure that the points are not the same and that the line is not vertical.

Q: What if I get a negative slope?

A: If you get a negative slope, it means that the line is sloping downward from left to right. This is a valid linear equation and can be written in slope-intercept form.

Q: Can I use the point-slope form of a linear equation?

A: Yes, you can use the point-slope form of a linear equation, which is y - y1 = m(x - x1), where (x1, y1) is one of the given points and m is the slope.

Q: How do I convert the point-slope form to slope-intercept form?

A: To convert the point-slope form to slope-intercept form, you can simplify the equation by distributing the slope (m) to the terms inside the parentheses and then combining like terms.

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

A: Yes, you can use the slope-intercept form to find the equation of a line that passes through a point and has a given slope. Simply plug in the values of the slope and the point into the equation y = mx + b and solve for b.

Q: What if I get a complex number as the slope?

A: If you get a complex number as the slope, it means that the line is not a real line and cannot be written in slope-intercept form. In this case, we cannot find a linear equation that satisfies the conditions.

Q: Can I use the equation of a line to find the equation of a circle?

A: No, you cannot use the equation of a line to find the equation of a circle. The equation of a circle is a different type of equation that is used to describe a circular region.

Q: Can I use the equation of a line to find the equation of a parabola?

A: No, you cannot use the equation of a line to find the equation of a parabola. The equation of a parabola is a different type of equation that is used to describe a parabolic region.

Conclusion

In conclusion, finding a linear equation satisfying the conditions given two points on the line is a straightforward process that involves finding the slope and y-intercept using the given points. We can then write the linear equation in slope-intercept form. By following the steps outlined in this article, we can answer some frequently asked questions related to this topic.

Tips and Tricks

  • Make sure to use the correct formula for the slope.
  • Make sure to use the correct point when finding the y-intercept.
  • Make sure to check if the line is vertical or not before writing the equation in slope-intercept form.
  • Make sure to simplify the equation by distributing the slope (m) to the terms inside the parentheses and then combining like terms.

Common Mistakes

  • Failing to use the correct formula for the slope.
  • Failing to use the correct point when finding the y-intercept.
  • Not checking if the line is vertical or not before writing the equation in slope-intercept form.
  • Not simplifying the equation by distributing the slope (m) to the terms inside the parentheses and then combining like terms.