Question

CRITICAL THINKING Consider the equation \(6 x+8 y=k .\) What numbers could replace \(k\) so that the \(x\) -intercept and the \(y\) -intercept are both integers? Explain.

Short answer

The values that could replace k so that the x-intercept and the y-intercept are both integers are the common multiples of 6 and 8, such as 24, 48, 72, and any other multiple of these numbers.

Step-by-step solution

Identify the given equation

The given equation is \(6x + 8y = k\). This is a linear equation in two variables.

Find the x-intercept

To find the x-intercept, set y = 0 in the equation and solve for x. The equation becomes \(6x + 8(0) = k\), which simplifies to \(6x = k\). Therefore, x would be \(x = \frac{k}{6}\). For x to be an integer, k must be a multiple of 6.

Find the y-intercept

To find the y-intercept, set x = 0 in the equation and solve for y. The equation becomes \(6(0) + 8y = k\), which simplifies to \(8y = k\). Therefore, y would be \(y = \frac{k}{8}\). For y to be an integer, k must be a multiple of 8.

Determine possible values of k

Based on step 2 and step 3, for both x and y to be integers, k must be a common multiple of 6 and 8. The common multiples of 6 and 8 are 24, 48, 72, and so on. So, the possible values for k are these common multiples.