A line has a slope of 3/4 and intersects the point (-12, -39

BTGmoderatorDC wrote:A line has a slope of 3/4 and intersects the point (-12, -39). At which point does this line intersect the x-axis?

A. (40,0)
B. (30,0)
C. (0,40)
D. (40,30)
E. (0,30)

TIP: If you were running short on time and encountered this question, you could quickly eliminate C, D and E and guess between A and B .
The coordinates of the x-intercept must be in the form (k, 0). So, we can eliminate C, D and E.

Now onto the solution...
Let's write the equation in slope y-intercept form: y = mx + b, where m is the slope and b is the y-intercept.
We know the slope is 3/4, so we have: y = (3/4)x + b

Next, since the point (-12, -39) lies ON the line, the coordinates x = -12 and y = -39 must satisfy the equation of the line.
Plugging those values into our equation, we get: -39 = (3/4)(-12) + b = 0
Simplify: -39 = -9 + b
Solve for b to get: b = -30

So, the equation of the given line is: y = (3/4)x - 30

We want to find the coordinates of the x-intercept.
This is the point where y = 0.
So, plug y = 0 into the equation to get: 0 = (3/4)x - 30 [now solve for x]
Add 30 to both sides to get: 30 = (3/4)x
Multiply both sides by 4/3 to get: 40 = x

So, the coordinates of the x-intercept are (40, 0)

Answer: A

Cheers,
Brent

BTGmoderatorDC wrote:A line has a slope of 3/4 and intersects the point (-12, -39). At which point does this line intersect the x-axis?

A. (40,0)
B. (30,0)
C. (0,40)
D. (40,30)
E. (0,30)

If we let a be the x-intercept of the line, then (a, 0) is a point on the line. Since the slope of the line is 3/4 and another point on the line is (-12, -39), using the slope formula, we have:

(0 - (-39))/(a - (-12)) = 3/4

39/(a + 12) = 3/4

3(a + 12) = 4(39)

3a + 36 = 156

3a = 120

a = 40

Alternate Solution:

Using the slope-intercept form of the equation of a line, we know that the equation of the line is y = (3/4)x + b for some number b. To find b, we substitute (-12, -39) into the equation:

-39 = (3/4)(-12) + b

-39 = -9 + b

b = -30

So, the equation of the line is y = (3/4)x - 30. To find the x-intercept, we simply plug in y = 0:

0 = (3/4)x - 30

(3/4)x = 30

x = 40

Thus, the line crosses the x-axis at the point (40, 0).

Answer: A