Question

4a. Explain how we can tell that this graph represents the give

Answer 1

a) the slope is 250

b)for every mile ( along thail) the hikers progress 250 meters ( elevation)

c)y=250x+500

3d)yes, the equation y-250x=500 represents the same relationship

Explanation

Step 1

slope of the line.

the slope of a line is given by the expression

$\begin{gathered} slope=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1} \\ \text{where} \\ P1(x_1,y_1)\text{ and} \\ P2(x_2,y_2) \\ are\text{ 2 points from the line} \end{gathered}$

then

pick up two poins from the line

$\begin{gathered} \text{let} \\ P1(0,500) \\ P2(2,1000) \end{gathered}$

replace in the expression

$\begin{gathered} slope=\frac{y_2-y_1}{x_2-x_1} \\ \text{slope}=\frac{1000-500}{2-0}=\frac{500}{2}=250 \end{gathered}$

so, the slope is 250

Step 2

what does the slope tell use about this situation

as the slope is rate of change

$\begin{gathered} slope=\frac{\Delta y}{\Delta x} \\ \text{replace the units ( check the a}\xi s) \\ \text{slope}=\text{ 250}\frac{ft}{\text{miles}} \end{gathered}$

so, the slope tell us

for every mile ( along thail) the hikers progress 250 meters ( elevation)

Step 3

equation:

to find the equation we can use the point slope equation

$\begin{gathered} y-y_1=m(x-x_1) \\ \text{where m is the slope} \\ \text{and} \\ P1(x_1,y_1) \end{gathered}$

replace

$\begin{gathered} y-y_1=m(x-x_1) \\ y-500=250(x-0) \\ y-500=250x \\ \text{add 500 in both sides} \\ y=250x+500 \end{gathered}$

Step 4

finally.

Does the equation y-250x=500 represent the same relationship?

if we start from

$y=250x+500$

then, we apply the subtraction property of equality ( which does not affect the function)

so

subtract 250 x in both sides

$\begin{gathered} y=250x+500 \\ \text{subtract 250 x in both sides} \\ y-250x=250x+500-250x \\ y-250x=500 \end{gathered}$

so

yes, the equation y-250x=500 represents the same relationship

I hope this helps you