Finding Other Points on the Line

Oct 13, 2025 by miodeya.com 90 views

Understanding the Problem

Vera wants to graph a line that passes through the point ${$ (0, 2)$}$ and has a slope of ${$ \frac{2}{3}$}$. To graph the line, Vera needs to find other points that lie on the line. In this article, we will explore the possible points that Vera can use to graph the line.

Recalling 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. In this case, the slope is ${$ \frac{2}{3}$}$ and the y-intercept is ${ $2\$}$ . Therefore, the equation of the line is ${$ y = \frac{2}{3}x + 2$}$.

Finding Other Points on the Line

To find other points on the line, we can substitute different values of ${$ x$}$ into the equation and solve for ${$ y$}$. Let's try substituting ${$ x = 3$}$ into the equation:

${$ y = \frac{2}{3}(3) + 2$}{ $y = 2 + 2\$}$ ${$ y = 4$}$

Therefore, the point ${$ (3, 4)$}$ lies on the line.

Checking Other Points

Now, let's try substituting ${$ x = -2$}$ into the equation:

${$ y = \frac{2}{3}(-2) + 2$}{ $y = -\frac{4}{3} + 2\$}$ ${$ y = \frac{2}{3}$}$

Therefore, the point ${$ (-2, \frac{2}{3})$}$ does not lie on the line.

Checking Another Point

Let's try substituting ${$ x = 6$}$ into the equation:

${$ y = \frac{2}{3}(6) + 2$}{ $y = 4 + 2\$}$ ${$ y = 6$}$

Therefore, the point ${$ (6, 6)$}$ lies on the line.

Conclusion

In conclusion, Vera can use the points ${$ (3, 4)$}$, ${$ (6, 6)$}$, and ${$ (0, 2)$}$ to graph the line. These points satisfy the equation of the line and lie on the line.

Answer Key

A. ${$ (3, 4)$}$ - Correct B. ${$ (-2, -3)$}$ - Incorrect C. ${$ (6, 6)$}$ - Correct

Discussion

Understanding the Problem

Q&A

Q: What is the equation of the line?

A: The equation of the line is ${$ y = \frac{2}{3}x + 2$}$.

Q: How can Vera find other points on the line?

A: Vera can substitute different values of ${$ x$}$ into the equation and solve for ${$ y$}$.

Q: What is the point ${$ (3, 4)$}$ on the line?

A: Yes, the point ${$ (3, 4)$}$ lies on the line. To verify this, Vera can substitute ${$ x = 3$}$ into the equation and solve for ${$ y$}$.

Q: What is the point ${$ (-2, -3)$}$ on the line?

A: No, the point ${$ (-2, -3)$}$ does not lie on the line. To verify this, Vera can substitute ${$ x = -2$}$ into the equation and solve for ${$ y$}$.

Q: What is the point ${$ (6, 6)$}$ on the line?

A: Yes, the point ${$ (6, 6)$}$ lies on the line. To verify this, Vera can substitute ${$ x = 6$}$ into the equation and solve for ${$ y$}$.

Q: How can Vera use the equation of the line to find other points on the line?

A: Vera can substitute different values of ${$ x$}$ into the equation and solve for ${$ y$}$. For example, Vera can substitute ${$ x = 9$}$ into the equation and solve for ${$ y$}$.

Q: What are some other ways to graph a line with a given slope and point?

A: There are several other ways to graph a line with a given slope and point, including:

Using the point-slope form of a line
Using the slope-intercept form of a line
Plotting two points on the line and drawing a line through them
Using a graphing calculator or computer software

Conclusion

In conclusion, Vera can use the points ${$ (3, 4)$}$, ${$ (6, 6)$}$, and ${$ (0, 2)$}$ to graph the line. These points satisfy the equation of the line and lie on the line. Vera can also use the equation of the line to find other points on the line by substituting different values of ${$ x$}$ into the equation and solving for ${$ y$}$.

Answer Key

A. ${$ (3, 4)$}$ - Correct B. ${$ (-2, -3)$}$ - Incorrect C. ${$ (6, 6)$}$ - Correct

Discussion

What other points could Vera use to graph the line? How can Vera use the equation of the line to find other points on the line? What are some other ways to graph a line with a given slope and point?