Question

The graph of a line in the cy-plane has a slope of 4 and contains the point( 1, 5). the graph of a second line passes through the points (0, 4) and(12, 0). if the two lines intersect at the point (a, b), what is the value ofa-b?

Answer 1

The equation of the first line is y = 4x + 1 and the second line is y = -1/3x + 4. Solving for their intersection point gives us (9/13, 49/13). Thus, a-b equals -40/13.

Step-by-step explanation:

To find the value of a-b where the two lines intersect, we need to determine the equations of both lines and then solve for their intersection point. The first line has a slope of 4 and contains the point (1, 5). Therefore, using the point-slope form, its equation can be written as y - y1 = m(x - x1), which simplifies to y - 5 = 4(x - 1) or y = 4x + 1.

The second line passes through the points (0, 4) and (12, 0). We can find its slope by using the slope formula m = (y2 - y1)/(x2 - x1). Thus, the slope of this line is (0 - 4)/(12 - 0) = -4/12 = -1/3. The y-intercept is 4, so its equation is y = -1/3x + 4.

To find the intersection point (a, b), we solve the two equations:
4x + 1 = -1/3x + 4.
Adding 1/3x to both sides and subtracting 1 yields:
4 1/3x = 3, which simplifies to
13/3x = 3. Therefore,
x = 3/(13/3), which simplifies to
x = 9/13, which is the value of a.

To find b, we substitute x = 9/13 into either equation, let's use the first one:
y = 4(9/13) + 1, which gives
y = 36/13 + 1, which can be written as
y = 36/13 + 13/13, simplifying to
y = 49/13.

Thus, the intersection point is (9/13, 49/13). The value of a-b is 9/13 - 49/13 = -40/13. So, a-b equals -40/13.

Answer 2

The point of intersection of the two lines is (9/13, 49/13), and the value of a - b is -40/13.

The point of intersection (a, b) of the two lines, we need to determine the equations of both lines.

For the first line with a slope of 4 and passing through the point (1, 5), we can use the point-slope form of a linear equation:

y - 5 = 4(x - 1).

y = 4x - 4 + 5, y = 4x + 1.

For the second line passing through the points (0, 4) and(12, 0), we can find the slope (m) and then use the point-slope form:

m = (y₂ - y₁)/(x₂ - x₁),

m = (0 - 4)/(12 - 0)

m = -1/3.

Now, using the point-slope form for the second line:

y - 4 = -1/3(x - 0).

y = -1/3x + 4.

Now, we can find the point of intersection by setting the two equations equal to each other:

4x + 1 = -1/3x + 4.

Solving for x: 13/3x = 3,

x = 9/13.

Substituting x back into either equation:

y = 4(9/13) + 1,

y = 49/13.

So, the point of intersection is (9/13, 49/13),and a - b = 9/13 - 49/13 = -40/13.