Question

graph the linear equation.

1. ( y=\frac{1}{2}x - 3 )
2. ( y=-\frac{x}{4}+1 )

find the slope of the line.
3.
4.

1. the cost ( y ) (in dollars) for ( x ) pounds of deli meat is represented by the equation ( y = 3.5x ). graph the equation and interpret the slope.
2. the amount ( y ) (in cups) of flour is proportional to the number ( x ) of eggs in a recipe. the recipe calls for 6 cups of flour for every 4 eggs.

a. write an equation that represents the situation.
b. interpret the slope.
c. how many eggs are included when the recipe uses 12 cups of flour?

Explanation:

Step1: Find the equation for part a

Since \(y\) (flour in cups) is proportional to \(x\) (number of eggs), the equation is of the form \(y = kx\). We know that when \(x = 4\), \(y=6\). Using the formula \(k=\frac{y}{x}\), we substitute the values: \(k=\frac{6}{4}=\frac{3}{2}\). So the equation is \(y=\frac{3}{2}x\).

Step2: Interpret the slope for part b

In the equation \(y = kx\) (a proportional relationship), the slope \(k\) represents the unit rate. Here, \(k = \frac{3}{2}\), which means that for every 1 egg (\(\Delta x=1\)), the amount of flour (\(y\)) increases by \(\frac{3}{2}\) cups. So the slope \(\frac{3}{2}\) is the number of cups of flour per egg.

Step3: Solve for \(x\) when \(y = 12\) in part c

We use the equation \(y=\frac{3}{2}x\). Substitute \(y = 12\) into the equation: \(12=\frac{3}{2}x\). To solve for \(x\), multiply both sides of the equation by \(\frac{2}{3}\). So \(x=12\times\frac{2}{3}\). \(12\times\frac{2}{3}=\frac{12\times2}{3}=8\).

Answer:

a. \(y=\frac{3}{2}x\)
b. The slope \(\frac{3}{2}\) means \(\frac{3}{2}\) cups of flour per egg.
c. \(8\) eggs.