Q. Find the \(x\)- and \(y\)-intercepts of the line \(x+3y=24.\)

Solution — find the x- and y-intercepts of the line

We are given the equation of a line:

\(x + 3y = 24\)

Recall the definitions:

• The x-intercept is the point where the line crosses the x-axis. At any point on the x-axis the y-coordinate is 0. To find the x-intercept, set \(y = 0\) and solve for \(x\).
• The y-intercept is the point where the line crosses the y-axis. At any point on the y-axis the x-coordinate is 0. To find the y-intercept, set \(x = 0\) and solve for \(y\).

1. Find the x-intercept (set \(y = 0\))
   1. Start with the equation: \(x + 3y = 24\).
   2. Substitute \(y = 0\): \(x + 3(0) = 24\).
   3. Simplify: \(x + 0 = 24\).
   4. Therefore \(x = 24\).
   5. The x-intercept is the point \((24, 0)\).
   6. Check: \(24 + 3(0) = 24\), which matches the given equation.
2. Find the y-intercept (set \(x = 0\))
   1. Start with the equation: \(x + 3y = 24\).
   2. Substitute \(x = 0\): \(0 + 3y = 24\).
   3. Simplify: \(3y = 24\).
   4. Divide both sides by 3: \(y = \dfrac{24}{3}\).
   5. Simplify: \(y = 8\).
   6. The y-intercept is the point \((0, 8)\).
   7. Check: \(0 + 3(8) = 24\), which matches the given equation.

Answer: x-intercept = (24, 0); y-intercept = (0, 8)