Question

exceeding: the money made from selling tacos is below, where x is the number of tacos sold and y is the amount of money made:
x | 4 | 5 | 6 | 7
y | 11 | 13 | 15 | 17
answer the following questions about the situation above:

• graph the table: graph grid
• why is this situation linear?
• what does the slope of the line mean in this context?
• if the slope were larger, what would that mean for the situation?

Explanation:

1. Graph the table:

To graph the table, we use the ordered pairs \((x, y)\) from the table: \((4, 11)\), \((5, 13)\), \((6, 15)\), \((7, 17)\).

• On a coordinate plane, the x - axis represents the number of tacos sold (\(x\)) and the y - axis represents the money made (\(y\)).
• Plot the points: For \(x = 4\), find \(y=11\) on the y - axis and mark the point \((4,11)\). Similarly, plot \((5,13)\), \((6,15)\) and \((7,17)\). Then, draw a straight line through these points (since we will see it is linear).

2. Why is this situation linear?

Step 1: Calculate the rate of change (slope) between consecutive points.

The formula for the slope \(m\) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).

For the points \((4,11)\) and \((5,13)\):
\(m_1=\frac{13 - 11}{5 - 4}=\frac{2}{1} = 2\)

For the points \((5,13)\) and \((6,15)\):
\(m_2=\frac{15 - 13}{6 - 5}=\frac{2}{1}=2\)

For the points \((6,15)\) and \((7,17)\):
\(m_3=\frac{17 - 15}{7 - 6}=\frac{2}{1}=2\)

Step 2: Analyze the rate of change.

Since the slope (rate of change) between any two consecutive points is constant (\(m = 2\) for all pairs of consecutive points), the relationship between the number of tacos sold (\(x\)) and the money made (\(y\)) is linear. In a linear relationship, the rate of change (slope) is constant, which means that as \(x\) increases by a fixed amount, \(y\) also increases by a fixed amount. Here, when \(x\) (number of tacos) increases by 1, \(y\) (money made) increases by 2.

Step 1: Recall the formula for slope in a real - world context.

The slope \(m=\frac{\text{Change in }y}{\text{Change in }x}\). In this context, \(x\) is the number of tacos sold and \(y\) is the money made.

Step 2: Interpret the slope.

We found that the slope \(m = 2\). The change in \(x\) is the change in the number of tacos sold, and the change in \(y\) is the change in the money made. So, a slope of 2 means that for each additional taco sold (an increase of 1 in \(x\)), the money made (\(y\)) increases by 2 units (the unit of \(y\) is the currency used for the money made, e.g., dollars, pesos, etc.).

Step 1: Recall the meaning of slope.

The slope represents the rate of change of \(y\) with respect to \(x\), i.e., \(\text{Slope}=\frac{\text{Change in money made}}{\text{Change in number of tacos sold}}\).

Step 2: Analyze the effect of a larger slope.

If the slope were larger, say the new slope is \(m' > 2\), then for a given change in the number of tacos sold (\(\Delta x\)), the change in the money made (\(\Delta y=m'\times\Delta x\)) would be larger. In practical terms, if the slope is larger, it means that for each taco sold, the amount of money made (the price per taco) is higher. For example, if originally we made $2 per taco (slope = 2), and the slope becomes 3, we would make $3 per taco. So, a larger slope implies that the profit (or revenue) per taco sold is greater.

Answer:

The situation is linear because the rate of change (slope) between consecutive \((x,y)\) pairs is constant. When \(x\) (tacos sold) increases by 1, \(y\) (money made) increases by 2 (the slope \(m = 2\) is constant for all consecutive points).

3. What does the slope of the line mean in this context?