Problem
A businesswoman buys a new computer for $\$ 3200$. For each year that she uses it, the value depreciates by $\$ 400$. The equation $y=-400 x+3200$ gives the value $y$ of the computer after $x$ years. What does the $x$-intercept mean in this situation? Find the $x$-intercept. After how many years will the value of the computer be $\$ 1600$ ? What does the $x$-intercept represent in this equation? A. The value $y$ of the computer when the number of years $x$ is zero. B. The number of years $\mathrm{x}$ when the value $y$ of the computer is zero. C. The number of years $x$ and the value $y$ of the computer are equal. The $\mathrm{x}$-intercept is (Type an ordered pair.) After years, the value of the computer will be $\$ 1600$.
Solution
Step 1 ：First, we need to find the x-intercept by setting y to 0 in the equation: \(y = -400x + 3200\)
Step 2 ：\(0 = -400x + 3200\)
Step 3 ：\(400x = 3200\)
Step 4 ：\(x = \frac{3200}{400}\)
Step 5 ：\(x = 8\)
Step 6 ：\(\boxed{(8, 0)}\) is the x-intercept.
Step 7 ：Next, we need to find the value of x when y is 1600: \(1600 = -400x + 3200\)
Step 8 ：\(400x = 1600\)
Step 9 ：\(x = \frac{1600}{400}\)
Step 10 ：\(x = 4\)
Step 11 ：After \(\boxed{4}\) years, the value of the computer will be $1600.
Step 12 ：The x-intercept represents option B: The number of years x when the value y of the computer is zero.
