Question

A student created this table to represent a linear relationship between x and y. X. Y-2. 10.0-1. 7.50. 5.01. 2.52. 0Write an equation of the line represented by the relationship between x and y shown in the table.

Answer 1

The equation of a line can be written using the slope-intercept form given as

$y=mx+c$

For that equation, we need to find m (rate of change) and c (y-intercept).

To find m, we can use the formula

$m=\frac{y_2-y_1}{x_2-x_1}$

Let us pick any two points on the graph such that

$\begin{gathered} (x_1,y_1)\Rightarrow(-2,10.0) \\ \text{and} \\ (x_2,y_2)\Rightarrow(1,2.5) \end{gathered}$

Substituting these into the equation, we have

$\begin{gathered} m=\frac{2.5-10.0}{1-(-2)} \\ m=-\frac{5}{2} \end{gathered}$

The intercept, c, can be calculated by substituting the rate of change, m, and any point coordinates to the equation of a line.

Let us pick the point

$(x,y)=(0,5.0)$

Substituting, we have

$\begin{gathered} 5=(-\frac{5}{2}\times0)+c \\ \therefore \\ c=5 \end{gathered}$

Therefore, we have the equation of the line to be

$y=-\frac{5}{2}x+5$

Multiplying all terms by 2 to clear out the fraction, we get the equation to be

$2y=-5x+10$