Question

the graph shows the relationship between the number of hours diego jogs (x) and the total distance he travels (y) in miles. the line passes through the point (3, 18).
which equation shows the relationship?
○ $y = 5x$
○ $y = 9x$
○ $y = 6x$
○ $y = 18x$

Explanation:

Step1: Recall the slope-intercept form

The equation of a line in slope - intercept form is $y = mx$, where $m$ is the slope (since the line passes through the origin, the $y$-intercept $b = 0$). The slope $m$ can be calculated using the formula $m=\frac{y_2 - y_1}{x_2 - x_1}$. We know that the line passes through the point $(x = 3,y = 18)$ and also through the origin $(0,0)$ (since the graph starts at $(0,0)$).

Step2: Calculate the slope

Using the formula $m=\frac{y_2 - y_1}{x_2 - x_1}$ with $(x_1,y_1)=(0,0)$ and $(x_2,y_2)=(3,18)$.
$m=\frac{18 - 0}{3 - 0}=\frac{18}{3}=6$

Step3: Write the equation

Since the slope $m = 6$ and the equation of the line is $y=mx$ (because it passes through the origin), the equation of the line is $y = 6x$. We can also verify by plugging $x = 3$ into each of the given options:

• For $y = 5x$, when $x = 3$, $y=5\times3 = 15

eq18$.

• For $y = 9x$, when $x = 3$, $y = 9\times3=27

eq18$.

• For $y = 6x$, when $x = 3$, $y=6\times3 = 18$, which matches the given point.
• For $y = 18x$, when $x = 3$, $y=18\times3 = 54

eq18$.

Answer:

$y = 6x$ (the option corresponding to $y = 6x$)