Question

the points (7,8) and (49,56) form a proportional relationship. find the slope of the line through the points. then use the slope to graph the line. (simplify your answer) which graph represents this relationship? a. graph b. graph c. graph

Explanation:

Step1: Recall slope formula

The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \).
Let \((x_1, y_1)=(7, 8)\) and \((x_2, y_2)=(49, 56)\).

Step2: Substitute values into formula

Substitute \( x_1 = 7 \), \( y_1 = 8 \), \( x_2 = 49 \), \( y_2 = 56 \) into the slope formula:
\( m=\frac{56 - 8}{49 - 7}=\frac{48}{42} \)

Step3: Simplify the fraction

Simplify \(\frac{48}{42}\) by dividing numerator and denominator by their greatest common divisor, which is 6:
\( \frac{48\div6}{42\div6}=\frac{8}{7} \)

For the graph part, we can check the slope of each graph. The slope of the line through \((7,8)\) and \((49,56)\) is \(\frac{8}{7}\), which means for every 7 units increase in \( x \), \( y \) increases by 8 units. Looking at the graphs, we can analyze the rise over run. But since we first found the slope as \(\frac{8}{7}\), and then we can match the graph with this slope.

Answer:

The slope is \(\frac{8}{7}\). To determine the graph, we check the slope of each option. The correct graph should have a slope of \(\frac{8}{7}\). (If we assume the graphs are labeled with axes, for example, graph A, B, C: we can check the points. For the line through \((7,8)\) and \((49,56)\), when \( x = 7 \), \( y = 8 \); \( x = 49 \), \( y = 56 \). So the graph should pass through these points or have the same slope. If we analyze the given graphs, let's assume graph A: check the slope. Wait, maybe the original problem's graphs: let's re - evaluate. The slope is \(\frac{8}{7}\), so the graph with slope \(\frac{8}{7}\) is the one where for \( x = 7 \), \( y = 8 \) and \( x = 49 \), \( y = 56 \). So the slope is \(\frac{8}{7}\), and the correct graph (assuming the options) would be the one that has this slope. But first, the slope calculation: \(\frac{8}{7}\).