Problem 88 - College Algebra

Question

Will help you prepare for the material covered in the next section. Consider the equation \(\frac{x^{2}}{16}-\frac{y^{2}}{9}=1\) a. Find the \(x\) -intercepts. b. Explain why there are no \(y\) -intercepts.

Step-by-Step Solution

Answer

The x-intercepts are (-4,0) and (4,0), and there are no y-intercepts for the given equation.

Step 1: Finding the x-intercepts

To find the x-intercepts, we need to set \(y = 0\), and solve for \(x\). So, our equation becomes: \(\frac{x^{2}}{16}-\frac{0^{2}}{9}=1\). This simplifies to \(\frac{x^{2}}{16}=1\), which can be further simplified to \(x^{2}=16\). Solving for \(x\), we obtain \(x= \pm 4\). Hence, the x-intercepts are (-4,0) and (4,0).

Step 2: Testing for y-intercepts

Next, we check for y-intercepts by setting \(x = 0\). This gives us \(\frac{0^{2}}{16}-\frac{y^{2}}{9}=1\), which simplifies to \(- \frac{y^{2}}{9} = 1\), after further simplification we get \(y^{2}=-9\). Since we cannot have a negative square, this equation has no real solutions, thus indicating that there are no y-intercepts for the given equation.