Question

.....................................................................................

Answer 1

(-6,0)

Explanation:

An equation of a line can be modeled as y = mx + b where m is slope and b is y-intercept.

For the line r, we can model the equation as y = mx + 3 since the line intersects y-axis at (0,3) as seen in the attachment.

For the line t, we can model the equation as y = mx - 6 as the problem gives y-intercept for line t equal to -6. Hence, the line t intersects y-axis at (0,6)

Next, we have to find the slope of line t by finding the slope of line r in the attachment. Apply the rise over run by counting the steps, you can see in the attachment that I put to learn how to count rise and run of a line. Also note that the value in attachment here is a scalar quantity, meaning only magnitude, no direction.

So we will have the slope of -1 since a line graph is heading down so the output decreases as input increases. Therefore, we know that m = -1 for both lines. Therefore, for the line t, we can model the new equation to:

`\displaystyle{y=-x-6}`

Then we find the x-intercept of the line by letting y = 0. Thus,

`\displaystyle{0=-x-6}\\\\\displaystyle{x=-6}`

Therefore, the x-intercept of line t is at (-6,0).

Answer 2

(-6,0)

Explanation:

We need to determine the equation of line r first to find the x-intercept of line t.

The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by the formula:

`slope = (y_2 - y_1)/(x_2 -x_1)`

For line r passing through (3, 0) and (0, 3), the slope is:

`slope = (3 - 0)/(0 - 3) = -1`

Since line t has the same slope as line r, its slope is also -1.

The equation of a line in slope-intercept form (y = mx + b) is determined by its slope (m) and y-intercept (b).

We know that the slope (m) of line t is -1, and the y-intercept (b) is -6. Substituting these values into the slope-intercept form, we get:

y = -x - 6

To find the x-intercept, we set y = 0 and solve for x:

0 = -x - 6

Adding x to both sides:

x = -6

Therefore, the x-intercept of line t is (-6,0).