The sum of the x- and y-intercepts of the graph of \(5x + 3y = k\) is 32.5.

Find the length of the line segment joining the x- and y-intercepts.

Sure you do.
What is the x-intercept ? It's the point where the graph crosses the x-axis. But y=0 at every point on the x-axis ! So the x-intercept is the point where
5x + 3(0) = k 5x = k x = 0.2 k
Similarly ... What is the y-intercept ? It's the point where the graph crosses the y-axis. But x=0 at every point on the y-axis ! So the y-intercept is the point where
5(0) + 3y = k 3y = k y = k/3
The sum of the x- and y-intercepts is 32.5 . 0.2k + k/3 = 32.5 Multiply each side by 5 : k + 5k/3 = 162.5 Multiply each side by 3 : 3k + 5k = 487.5 8k = 487.5 Divide each side by 8 : k = 60.9375
The x-intercept is (0.2k, 0) = the point (12.1875, 0) .
The y-intercept is (0, k/3) = the point (0, 20.3125) .
Those are the two points. Now it wants you to find the distance between them. Do you remember how to find the distance between 2 points ?
Find the (difference of their x-values) and square it. Find the (difference of their y-values) and square it. Add the two squares together. Take the square root of the sum. That's the distance between the points.
(12.1875)² + (20.3125)² = the square of the distance.
If you don't get 23.688 for the distance, then check your arithmetic.

Answered by AL2006 | 2024-06-10

The length of the line segment joining the x- and y-intercepts of the line 5 x + 3 y = k is approximately 23.75 units. We found the intercepts, calculated their coordinates based on k, and then used the distance formula to find the length of the segment connecting those two points.
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Answered by AL2006 | 2024-12-26