Solution: Write Equations Using Point-Slope
- Use the Point-Slope formula to write the equation of the line between the points \((0,2)\) and \((-2, -1)\).
\[slope=\frac{-1-(2)}{-2-(0)}=\frac{-3}{-2}=\frac{3}{2}\]
\[\begin{align}y-2&=\dfrac{3}{2}(x-0)\\\\ y-2&=\frac{3}{2}x\\\\ y&=\dfrac{3}{2}x+2\end{align}\]
- To check if a relationship is linear, we would need to see that there is a constant rate of change between any of the values in the table. We do this by evaluating the change in \(x\) and the change in \(y\).
| Number of cups in the stack | Height of the stack (inches) |
| \(3\) | \(2\dfrac{3}{8}\) |
| \(6\) | \(2\dfrac{3}{4}\) |
| \(9\) | \(3\dfrac{1}{8}\) |
| \(12\) | \(3\dfrac{1}{2}\) |
- Since the domain (\(x\)-values) increase by the constant 3, we next determine if the range (\(y\)-values) also increase or decrease by a constant amount.
- The change between \(2\dfrac{3}{8}\) and \(2\dfrac{3}{4}\) is \(\dfrac{3}{8}\).
- The change between \(2\dfrac{3}{4}\) and \(3\dfrac{1}{8}\) is \(\dfrac{3}{8}\).
- The change between \(3\dfrac{1}{8}\) and \(3\dfrac{1}{2}\) is \(\dfrac{3}{8}\).
Since the \(y\)-values are increasing by \(\dfrac{3}{8}\) and the \(x\)-values are increasing by 3, the relationship is linear.
- By selecting two points from the table \(\bigg(3, 2\dfrac{3}{8}\bigg)\) and \(\bigg(9, 3\dfrac{1}{8}\bigg)\), we can calculate the slope as \(\Bigg(\dfrac{3\dfrac{1}{8}-2\dfrac{3}{8}}{9-3}\Bigg)\) = \(\dfrac{1}{8}\)
Use point slope to find the equation: \(\bigg(y-2\dfrac{3}{8}\bigg) = \dfrac{1}{8}(x-3)\)
Equation: \(y = \dfrac{1}{8}x+2\)
- The equation tells us that the height of the stack grows by \(\dfrac{1}{8}\) inches for every cup added, and that the height of the shelf is 2 inches.
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