lines a and b are perpendicular and have the same x-intercept. The equation of line a is x+2y-4=0. Determine the y-intercept of line b.
line a has slope -1/2, and goes through (4,0)
line b has slope 2 and goes through (4,0):
2x - y - 8 = 0
To determine the y-intercept of line b, we need to first find the slope of line a.
The equation of line a is given as x + 2y - 4 = 0.
To find the slope, we need to rearrange the equation into slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.
Step 1: Move the x term to the other side of the equation:
2y = -x + 4
Step 2: Divide both sides by 2 to isolate y:
y = (-1/2)x + 2
Now we can conclude that the slope of line a is -1/2.
Since lines a and b are perpendicular, their slopes are negative reciprocals of each other.
The negative reciprocal of -1/2 is 2/1 or simply 2.
Now that we know the slope of line b is 2, we can use the slope-intercept form to find the equation of line b.
Since lines a and b have the same x-intercept, we know that the x-coordinate of the x-intercept is the same for both lines. In other words, the x-coordinate at which both lines intersect is the same.
For line a, the x-intercept occurs when y = 0. Substituting y with 0 in the equation of line a, we can solve for x:
x + 2(0) - 4 = 0
x - 4 = 0
x = 4
So the x-intercept for both lines a and b is x = 4.
Now we can use the slope-intercept form (y = mx + b) and substitute x = 4 to find the y-intercept of line b:
y = 2(4) + b
y = 8 + b
Since line b has the same x-intercept as line a, the x-coordinate is 4. Therefore, when x = 4, y should also be equal to 0 for line b.
So we can set y = 0 in the equation above and solve for b:
0 = 8 + b
b = -8
Therefore, the y-intercept of line b is -8.