## Solution: Write Equations Using Point-Slope

- Use the
**Point-Slope****formula**

\[slope=\frac{-1-(2)}{-2-(0)}=\frac{-3}{-2}=\frac{3}{2}\]

\[\begin{align}y-2&=\dfrac{3}{2}(x-0)\\\\ y-2&=\frac{3}{2}x\\\\ y&=\dfrac{3}{2}x+2\end{align}\]

- 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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