Final answer:
The y-intercept of the line perpendicular to y = (1/2)x - 6 and passing through (3,5) is 11. This is found by determining the slope of the new line as -2 and using the point-slope form to find the line equation, and then identifying the y-intercept from this equation.
Step-by-step explanation:
To find the y-intercept of a line that is perpendicular to the given line y = (1/2)x - 6 and passes through the point (3,5), first, determine the slope of the given line. Since the slope of the given line is 1/2, the slope of the line perpendicular to it will be the negative reciprocal, which is -2. Using this slope and the point (3,5), we can apply the point-slope form of the equation of a line, y - y1 = m(x - x1), where m is the slope and (x1, y1) is the point it passes through.
The equation of the line perpendicular to the given line and passing through (3,5) is: y - 5 = -2(x - 3). Simplifying this, we get y = -2x + 11. Now that we have the equation of the perpendicular line, we can find its y-intercept by setting x to be 0. The y-intercept is therefore the value of y when x equals zero, which gives us y = 11. Thus, the y-intercept of the perpendicular line is 11.
The given line has a slope of 1/2, so the perpendicular line will have a slope that is the negative reciprocal of 1/2, which is -2.
Next, we use the point-slope form of a line, y - y1 = m(x - x1), where (x1, y1) is the given point (3,5) and m is the slope. Plugging in the values, we get y - 5 = -2(x - 3).
Simplifying the equation, we get y - 5 = -2x + 6. Now, we can isolate y by adding 5 to both sides of the equation. The final equation of the line is y = -2x + 11.
Therefore, the y-intercept of the line perpendicular to y = (1/2)x - 6 and passes through (3,5) is 11.