Question

a familys dishwasher uses the same number of gallons of water for each load. the table shows how many gallons of water, g, the dishwasher uses for n loads of dishes. what equation can you use to represent the relationship shown in the table? show your work. dishwasher water use number of loads, n gallons used, g 4 18.0 5 22.5 6 27.0 7 31.5

Explanation:

Step1: Find the rate of water use per load

To find the rate, we can use the formula for slope (rate of change) between two points \((n_1, g_1)\) and \((n_2, g_2)\), which is \(m=\frac{g_2 - g_1}{n_2 - n_1}\). Let's take the first two points \((4, 18.0)\) and \((5, 22.5)\).
\(m=\frac{22.5 - 18.0}{5 - 4}=\frac{4.5}{1} = 4.5\)
We can check with another pair, say \((5, 22.5)\) and \((6, 27.0)\):
\(m=\frac{27.0 - 22.5}{6 - 5}=\frac{4.5}{1}=4.5\)
So the rate of water use per load is \(4.5\) gallons per load.

Step2: Determine the equation of the line

Since the relationship is linear (constant rate), we use the slope - intercept form of a linear equation \(g=mn + b\). But since when \(n = 0\), \(g = 0\) (if there are 0 loads, 0 gallons are used), the \(y\) - intercept \(b = 0\). So the equation is \(g=4.5n\) (or \(g=\frac{9}{2}n\) or \(g = 4.5n\)).

Answer:

The equation representing the relationship is \(g = 4.5n\) (or equivalent forms like \(g=\frac{9}{2}n\)).