Question

a proportional relationship is shown in the table below:
x 0 1.3 2.6 3.9 5.2
y 0 1 2 3 4
what is the slope of the line that represents this relationship?
graph the line that represents this relationship.

Explanation:

Step1: Recall slope formula

The slope \( m \) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \).

Step2: Choose two points from the table

Let's take the points \((1.3, 1)\) and \((2.6, 2)\) (we could also use other pairs like \((0,0)\) and \((1.3,1)\) etc.).

Step3: Substitute into slope formula

Substitute \( x_1 = 1.3 \), \( y_1 = 1 \), \( x_2 = 2.6 \), \( y_2 = 2 \) into the formula:
\( m=\frac{2 - 1}{2.6 - 1.3}=\frac{1}{1.3}=\frac{10}{13}\approx0.769 \)? Wait, no, wait. Wait, let's check with \((0,0)\) and \((1.3,1)\): \( m=\frac{1 - 0}{1.3 - 0}=\frac{1}{1.3}=\frac{10}{13}\)? Wait, no, wait, 1 divided by 1.3 is equal to \( \frac{10}{13}\approx0.769 \)? Wait, no, wait, 1.3 times 1 is 1.3, 1.3 times 2 is 2.6, 1.3 times 3 is 3.9, 1.3 times 4 is 5.2. So the relationship is \( y=\frac{1}{1.3}x \)? Wait, no, actually, if we take \( x = 1.3 \), \( y = 1 \), so \( y=\frac{1}{1.3}x=\frac{10}{13}x \)? Wait, no, wait, 1 divided by 1.3 is the same as \( \frac{10}{13}\approx0.769 \)? Wait, no, wait, 1.3 is \( \frac{13}{10} \), so \( \frac{1}{1.3}=\frac{10}{13}\approx0.769 \)? Wait, but let's do it with two points. Let's take \((x_1,y_1)=(0,0)\) and \((x_2,y_2)=(5.2,4)\). Then \( m=\frac{4 - 0}{5.2 - 0}=\frac{4}{5.2}=\frac{40}{52}=\frac{10}{13}\approx0.769 \)? Wait, no, 5.2 divided by 4 is 1.3, so 4 divided by 5.2 is \( \frac{4}{5.2}=\frac{40}{52}=\frac{10}{13}\approx0.769 \)? Wait, no, wait, 1.3 times 4 is 5.2, so the slope is \( \frac{1}{1.3}=\frac{10}{13}\approx0.769 \)? Wait, no, wait, slope is rise over run, so \( \frac{\Delta y}{\Delta x} \). Let's take \( \Delta y = 1 \), \( \Delta x = 1.3 \), so slope is \( \frac{1}{1.3}=\frac{10}{13}\approx0.769 \)? Wait, no, wait, 1.3 is \( \frac{13}{10} \), so \( \frac{1}{\frac{13}{10}}=\frac{10}{13}\approx0.769 \). But actually, 1 divided by 1.3 is equal to \( \frac{10}{13}\approx0.769 \). Wait, but let's check with another pair. Take \((1.3,1)\) and \((2.6,2)\). \( \Delta y = 2 - 1 = 1 \), \( \Delta x = 2.6 - 1.3 = 1.3 \), so \( m=\frac{1}{1.3}=\frac{10}{13}\approx0.769 \). Alternatively, since it's a proportional relationship, the slope is \( \frac{y}{x} \) for any non - zero \( x \). So for \( x = 1.3 \), \( y = 1 \), so slope \( m=\frac{y}{x}=\frac{1}{1.3}=\frac{10}{13}\approx0.769 \). Wait, but 1.3 is \( \frac{13}{10} \), so \( \frac{1}{\frac{13}{10}}=\frac{10}{13}\approx0.769 \). But let's simplify \( \frac{4}{5.2} \): divide numerator and denominator by 0.4, we get \( \frac{10}{13}\approx0.769 \). So the slope is \( \frac{10}{13}\) or approximately 0.769. Wait, but actually, 1.3 is \( \frac{13}{10} \), so the slope is \( \frac{10}{13}\approx0.769 \). But let's check: if \( x = 1.3 \), \( y = 1 \), so \( y=\frac{10}{13}x \), when \( x = 2.6 \), \( y=\frac{10}{13}\times2.6=\frac{10}{13}\times\frac{26}{10}=2 \), which matches. When \( x = 3.9 \), \( y=\frac{10}{13}\times3.9=\frac{10}{13}\times\frac{39}{10}=3 \), which matches. When \( x = 5.2 \), \( y=\frac{10}{13}\times5.2=\frac{10}{13}\times\frac{52}{10}=4 \), which matches. So the slope is \( \frac{10}{13}\approx0.769 \), but in fraction form, \( \frac{10}{13}\) or we can write it as \( \frac{1}{1.3}=\frac{10}{13}\). Wait, but 1.3 is \( \frac{13}{10} \), so the reciprocal is \( \frac{10}{13}\). So the slope is \( \frac{10}{13}\) or approximately 0.769. But let's do it with two points: \((1.3,1)\) and \((2.6,2)\). \( m=\frac{2 - 1}{2.6 - 1.3}=\frac{1}{1.3}=\frac{10}{13}\approx0.769 \). So the slope is \( \frac{10}{13}\) (or approximately 0.77, but as a fraction, \( \frac{10}{13}\)).

Wait, no, wait, 1.3 is equal to \( \frac{13}{10} \), so \( \frac{1}{1.3}=\frac{10}{13}\approx0.769 \). So the slope is \( \frac{10}{13}\) or approximately 0.77. But let's check with another pair: \((0,0)\) and \((1.3,1)\). \( m=\frac{1 - 0}{1.3 - 0}=\frac{1}{1.3}=\frac{10}{13}\). So that's the slope.

Answer:

\(\frac{10}{13}\) (or approximately \(0.77\))