What is the length of the segment of the line 3x + 4y = 12 between the x-axis and y-axis?

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Q: What is the length of the segment of the line 3x + 4y = 12 between the x-axis and y-axis?

Solution: The x-intercept is (4, 0) and the y-intercept is (0, 3). The length of the segment is sqrt((4-0)^2 + (0-3)^2) = sqrt(16 + 9) = 5.

Steps: 10

Step 1: Identify the equation of the line, which is 3x + 4y = 12.
Step 2: Find the x-intercept by setting y = 0 in the equation. This gives us 3x + 4(0) = 12, which simplifies to 3x = 12. Solving for x gives x = 4. So, the x-intercept is (4, 0).
Step 3: Find the y-intercept by setting x = 0 in the equation. This gives us 3(0) + 4y = 12, which simplifies to 4y = 12. Solving for y gives y = 3. So, the y-intercept is (0, 3).
Step 4: Now we have the two points: the x-intercept (4, 0) and the y-intercept (0, 3).
Step 5: Use the distance formula to find the length of the segment between these two points. The formula is length = sqrt((x2 - x1)^2 + (y2 - y1)^2). Here, (x1, y1) = (4, 0) and (x2, y2) = (0, 3).
Step 6: Substitute the values into the formula: length = sqrt((0 - 4)^2 + (3 - 0)^2).
Step 7: Calculate (0 - 4)^2 = 16 and (3 - 0)^2 = 9.
Step 8: Add these results: 16 + 9 = 25.
Step 9: Take the square root of 25, which is 5.
Step 10: Therefore, the length of the segment of the line between the x-axis and y-axis is 5.