What Is The Equation That Models The Line Between The Points $(-6,7)$ And $(15,-7)$?1. Calculate The Slope ( M (m ( M ]:$m = \frac{y_2-y_1}{x_2-x_1}$2. Use The Slope-intercept Form Y = M X + B Y = Mx + B Y = M X + B To Find The

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What is the Equation that Models the Line between the Points (-6,7) and (15,-7)?

Understanding the Problem

The problem requires finding the equation of a line that passes through two given points, (-6,7) and (15,-7). To solve this problem, we need to use the concept of slope and the slope-intercept form of a line.

Calculating the Slope

The slope of a line is a measure of how steep it is. It can be calculated using the formula:

m = \frac{y_2-y_1}{x_2-x_1}

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

In this case, the coordinates of the two points are (-6,7) and (15,-7). Plugging these values into the formula, we get:

m = \frac{-7-7}{15-(-6)} m = \frac{-14}{21} m = -\frac{2}{3}

Using the Slope-Intercept Form

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.

We have already calculated the slope (m) as -\frac{2}{3}. Now, we need to find the value of b.

To do this, we can use one of the given points, say (-6,7). Plugging these values into the equation, we get:

7 = -\frac{2}{3}(-6) + b 7 = 4 + b b = 3

Finding the Equation of the Line

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

y = -\frac{2}{3}x + 3

Interpreting the Equation

The equation y = -\frac{2}{3}x + 3 represents a line with a slope of -\frac{2}{3} and a y-intercept of 3. This means that for every 3 units we move to the right, the line drops by 2 units.

Graphing the Line

To graph the line, we can use the slope-intercept form of the equation. We can start by plotting the y-intercept (0,3) on the coordinate plane. Then, we can use the slope to find another point on the line. Since the slope is -\frac{2}{3}, we can move 3 units to the right and 2 units down from the y-intercept to find another point on the line.

Conclusion

In this article, we have learned how to find the equation of a line that passes through two given points. We have used the concept of slope and the slope-intercept form of a line to solve the problem. The equation of the line is y = -\frac{2}{3}x + 3.

Key Takeaways

  • The slope of a line can be calculated using the formula m = \frac{y_2-y_1}{x_2-x_1}.
  • The slope-intercept form of a line is given by the equation y = mx + b.
  • The equation of a line can be written in slope-intercept form using the values of m and b.

Frequently Asked Questions

  • Q: What is the slope of the line that passes through the points (-6,7) and (15,-7)? A: The slope of the line is -\frac{2}{3}.
  • Q: What is the y-intercept of the line that passes through the points (-6,7) and (15,-7)? A: The y-intercept of the line is 3.
  • Q: What is the equation of the line that passes through the points (-6,7) and (15,-7)? A: The equation of the line is y = -\frac{2}{3}x + 3.

References

Additional Resources

Introduction

In our previous article, we learned how to find the equation of a line that passes through two given points. We used the concept of slope and the slope-intercept form of a line to solve the problem. In this article, we will answer some frequently asked questions related to the equation of a line.

Q: What is the slope of the line that passes through the points (-6,7) and (15,-7)?

A: The slope of the line is -\frac{2}{3}. To calculate the slope, we use the formula m = \frac{y_2-y_1}{x_2-x_1}, where (x1, y1) and (x2, y2) are the coordinates of the two points.

Q: What is the y-intercept of the line that passes through the points (-6,7) and (15,-7)?

A: The y-intercept of the line is 3. To find the y-intercept, we can use the slope-intercept form of the equation, y = mx + b, and plug in the values of m and one of the given points.

Q: What is the equation of the line that passes through the points (-6,7) and (15,-7)?

A: The equation of the line is y = -\frac{2}{3}x + 3. This equation represents a line with a slope of -\frac{2}{3} and a y-intercept of 3.

Q: How do I graph the line on a coordinate plane?

A: To graph the line, we can use the slope-intercept form of the equation. We can start by plotting the y-intercept (0,3) on the coordinate plane. Then, we can use the slope to find another point on the line. Since the slope is -\frac{2}{3}, we can move 3 units to the right and 2 units down from the y-intercept to find another point on the line.

Q: What is the significance of the slope in the equation of a line?

A: The slope of a line represents the rate of change of the line. It tells us how steep the line is and how much it rises or falls for every unit we move to the right.

Q: Can I use the equation of a line to find the coordinates of a point on the line?

A: Yes, you can use the equation of a line to find the coordinates of a point on the line. To do this, you can plug in the x-coordinate of the point into the equation and solve for the y-coordinate.

Q: How do I determine if a point lies on a line?

A: To determine if a point lies on a line, you can plug the coordinates of the point into the equation of the line and check if the equation is true.

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

A: Yes, you can use the equation of a line to find the equation of a parallel line. To do this, you can use the fact that parallel lines have the same slope.

Q: How do I find the equation of a line that is perpendicular to a given line?

A: To find the equation of a line that is perpendicular to a given line, you can use the fact that perpendicular lines have slopes that are negative reciprocals of each other.

Conclusion

In this article, we have answered some frequently asked questions related to the equation of a line. We have discussed the significance of the slope in the equation of a line, how to graph a line on a coordinate plane, and how to find the equation of a line that is parallel or perpendicular to a given line.

Key Takeaways

  • The slope of a line represents the rate of change of the line.
  • The equation of a line can be used to find the coordinates of a point on the line.
  • The equation of a line can be used to determine if a point lies on the line.
  • Parallel lines have the same slope.
  • Perpendicular lines have slopes that are negative reciprocals of each other.

Frequently Asked Questions

  • Q: What is the slope of the line that passes through the points (-6,7) and (15,-7)? A: The slope of the line is -\frac{2}{3}.
  • Q: What is the y-intercept of the line that passes through the points (-6,7) and (15,-7)? A: The y-intercept of the line is 3.
  • Q: What is the equation of the line that passes through the points (-6,7) and (15,-7)? A: The equation of the line is y = -\frac{2}{3}x + 3.

References

Additional Resources