What Is The $y$-intercept Of The Line In The Standard $(x, Y)$ Coordinate Plane That Goes Through The Points $(-3, 6)$ And $(3, 2)$?F. 0 G. 2 H. 4 J. 6 K. 8

Understanding the Problem

To find the $y$-intercept of a line in the standard $(x, y)$ coordinate plane, we need to understand the concept of the $y$-intercept. The $y$-intercept is the point at which the line intersects the $y$-axis. In other words, it is the value of $y$ when $x = 0$.

Finding the Slope of the Line

The first step in finding the $y$-intercept of the line is to find the slope of the line. The slope of a line is a measure of how steep the line is. It can be calculated using the formula:

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

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

In this case, we are given two points: $(-3, 6)$ and $(3, 2)$. We can use these points to calculate the slope of the line.

Calculating the Slope

Using the formula for slope, we get:

$$m = \frac{2 - 6}{3 - (-3)}$$

$$m = \frac{-4}{6}$$

$$m = -\frac{2}{3}$$

Finding the Equation of the Line

Now that we have the slope of the line, we can use it to find the equation of the line. The equation of a line can be written in the form:

$$y = mx + b$$

where $m$ is the slope of the line and $b$ is the $y$-intercept.

We already know the slope of the line, which is $-\frac{2}{3}$. We can use one of the points to find the value of $b$.

Using a Point to Find the Value of $b$

We can use the point $(3, 2)$ to find the value of $b$. Plugging in the values of $x$, $y$, and $m$, we get:

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

$$2 = -2 + b$$

$$b = 4$$

Writing the Equation of the Line

Now that we have the value of $b$, we can write the equation of the line:

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

Finding the $y$-Intercept

The $y$-intercept of the line is the value of $y$ when $x = 0$. We can find this value by plugging in $x = 0$ into the equation of the line:

$$y = -\frac{2}{3}(0) + 4$$

$$y = 4$$

Conclusion

The $y$-intercept of the line in the standard $(x, y)$ coordinate plane that goes through the points $(-3, 6)$ and $(3, 2)$ is $4$. This means that the line intersects the $y$-axis at the point $(0, 4)$.

Final Answer

The final answer is $\boxed{4}$.

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. In other words, it is the value of $y$ when $x = 0$.

Q: How do I find the $y$-intercept of a line?

A: To find the $y$-intercept of a line, you need to find the equation of the line. You can use the slope-intercept form of a line, which is $y = mx + b$, where $m$ is the slope of the line and $b$ is the $y$-intercept.

Q: What is the slope of a line?

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

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

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

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

A: To find the slope of a line using two points, you can use the formula:

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

For example, if you have the points $(2, 3)$ and $(4, 5)$, you can plug in the values to get:

$$m = \frac{5 - 3}{4 - 2}$$

$$m = \frac{2}{2}$$

$$m = 1$$

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

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

Q: How do I find the $y$-intercept of a line using the equation of the line?

A: To find the $y$-intercept of a line using the equation of the line, you can plug in $x = 0$ into the equation. For example, if the equation of the line is $y = 2x + 3$, you can plug in $x = 0$ to get:

$$y = 2(0) + 3$$

$$y = 3$$

Q: What is the $y$-intercept of the line in the standard $(x, y)$ coordinate plane that goes through the points $(-3, 6)$ and $(3, 2)$?

A: The $y$-intercept of the line in the standard $(x, y)$ coordinate plane that goes through the points $(-3, 6)$ and $(3, 2)$ is $4$. This means that the line intersects the $y$-axis at the point $(0, 4)$.

Q: How do I find the $y$-intercept of a line using a graph?

A: To find the $y$-intercept of a line using a graph, you can look for the point where the line intersects the $y$-axis. This point will have an $x$-coordinate of $0$ and a $y$-coordinate that is the $y$-intercept of the line.

Q: What is the difference between the $y$-intercept and the $x$-intercept of a line?

A: The $y$-intercept of a line is the point where the line intersects the $y$-axis, while the $x$-intercept of a line is the point where the line intersects the $x$-axis. The $y$-intercept is the value of $y$ when $x = 0$, while the $x$-intercept is the value of $x$ when $y = 0$.

Q: How do I find the $x$-intercept of a line?

A: To find the $x$-intercept of a line, you can plug in $y = 0$ into the equation of the line. For example, if the equation of the line is $y = 2x + 3$, you can plug in $y = 0$ to get:

$$0 = 2x + 3$$

$$-3 = 2x$$

$$x = -\frac{3}{2}$$

Q: What is the $x$-intercept of the line in the standard $(x, y)$ coordinate plane that goes through the points $(-3, 6)$ and $(3, 2)$?

A: The $x$-intercept of the line in the standard $(x, y)$ coordinate plane that goes through the points $(-3, 6)$ and $(3, 2)$ is $-3$. This means that the line intersects the $x$-axis at the point $(-3, 0)$.

Q: How do I use the $y$-intercept and the $x$-intercept to graph a line?

A: To graph a line using the $y$-intercept and the $x$-intercept, you can start by plotting the $y$-intercept on the graph. Then, you can use the $x$-intercept to find the point where the line intersects the $x$-axis. Finally, you can draw a line through the two points to graph the line.

Q: What is the relationship between the $y$-intercept and the $x$-intercept of a line?

A: The $y$-intercept and the $x$-intercept of a line are related in that they are both points on the line. The $y$-intercept is the point where the line intersects the $y$-axis, while the $x$-intercept is the point where the line intersects the $x$-axis.