Identify The Slope And Point From The Following Line.$y - 9 = 6(x - 5$\]Slope: \_\_\_\_ Point: \_\_\_\_

Feb 27, 2025 by ADMIN 104 views

**Slope and Point Identification: A Step-by-Step Guide**

Understanding the Basics of Slope and Point

In mathematics, the slope of a line is a measure of how steep it is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. The point, on the other hand, is a specific location on the line, represented by an ordered pair of coordinates (x, y).

Identifying the Slope and Point from a Given Equation

To identify the slope and point from a given equation, we need to follow a step-by-step approach.

Step 1: Rewrite the Equation in Slope-Intercept Form

The given equation is $y - 9 = 6(x - 5)$ . To rewrite it in slope-intercept form, we need to expand the right-hand side and simplify the equation.

y - 9 = 6x - 30
y = 6x - 21

Step 2: Identify the Slope

The slope-intercept form of a linear equation is $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept. In our rewritten equation, $y = 6x - 21$ , the slope is the coefficient of $x$ , which is $6$ .

Step 3: Identify the Point

The point is the location on the line where the line intersects the y-axis. In the slope-intercept form, the y-intercept is the value of $b$ . In our rewritten equation, the y-intercept is $-21$ . Therefore, the point is $(0, -21)$ .

Conclusion

In conclusion, the slope of the line is $6$ and the point is $(0, -21)$ .

Example Use Case

Suppose we have a line with a slope of $4$ and a point of $(2, 3)$ . We can use the point-slope form of a linear equation to write the equation of the line.

y - 3 = 4(x - 2)
y - 3 = 4x - 8
y = 4x - 5

Tips and Tricks

To identify the slope and point from a given equation, rewrite the equation in slope-intercept form.
The slope is the coefficient of $x$ in the slope-intercept form.
The point is the location on the line where the line intersects the y-axis, represented by the y-intercept in the slope-intercept form.

Common Mistakes to Avoid

Not rewriting the equation in slope-intercept form.
Not identifying the slope and point correctly.
Not using the correct formula to calculate the slope and point.

Conclusion

Frequently Asked Questions

Q: What is the slope of a line?

A: The slope of a line is a measure of how steep it is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.

Q: How do I identify the slope from a given equation?

A: To identify the slope from a given equation, rewrite the equation in slope-intercept form. The slope is the coefficient of $x$ in the slope-intercept form.

Q: What is the point of a line?

A: The point of a line is a specific location on the line, represented by an ordered pair of coordinates (x, y).

Q: How do I identify the point from a given equation?

A: To identify the point from a given equation, rewrite the equation in slope-intercept form. The point is the location on the line where the line intersects the y-axis, represented by the y-intercept in the slope-intercept form.

Q: What is the difference between the slope and the point?

A: The slope is a measure of how steep the line is, while the point is a specific location on the line.

Q: Can I use the point-slope form to identify the slope and point?

A: Yes, you can use the point-slope form to identify the slope and point. However, it is often easier to rewrite the equation in slope-intercept form.

Q: What is the formula for calculating the slope?

A: The formula for calculating the slope is:

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

where $m$ is the slope, and $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the line.

Q: What is the formula for calculating the point?

A: The formula for calculating the point is:

(x, y) = (0, b)

where $(x, y)$ is the point, and $b$ is the y-intercept.

Q: Can I use a graphing calculator to identify the slope and point?

A: Yes, you can use a graphing calculator to identify the slope and point. However, it is often easier to use the formulas and equations outlined in this article.

Q: What are some common mistakes to avoid when identifying the slope and point?

A: Some common mistakes to avoid when identifying the slope and point include:

Not rewriting the equation in slope-intercept form.
Not identifying the slope and point correctly.
Not using the correct formula to calculate the slope and point.

Conclusion

In conclusion, identifying the slope and point from a given equation is a crucial skill in mathematics. By following the step-by-step approach outlined in this article, you can confidently identify the slope and point from any given equation. Remember to rewrite the equation in slope-intercept form, identify the slope and point correctly, and use the correct formula to calculate the slope and point.

Additional Resources

Mathway: A online math problem solver that can help you identify the slope and point from a given equation.
Khan Academy: A free online learning platform that offers video lessons and practice exercises on mathematics, including slope and point identification.
Wolfram Alpha: A online calculator that can help you identify the slope and point from a given equation.