Q41. - Algebra 2

Question

Use the table that shows the number of people below the poverty level for the period 1980-1998.

 41. Use two ordered pairs to write a prediction equation 

Step-by-Step Solution

Answer

The prediction equation for the data is $y=0.76x+29.3$.

Step-1 – Apply the concept of slope-intercept form

Equation of line in slope intercept form is expressed below.

$y=mx+c$

Where m is the slope and c is the intercept of y-axis.

Step-2 – Apply the concept of slope

Consider the tabulated data for number of people below the poverty level for the period 1980-1998.

$\begin{array}{cc}\text{Year}& \begin{array}{l}\text{People}\\ \left(\text{millions}\right)\end{array}\\ 1980& 29.3\\ 1985& 33.1\\ 1990& 33.6\\ 1995& 36.4\\ 1998& 34.5\end{array}$

The ordered pairs that represent number of people in the given year are provided below.

Since, the years starts from 1980, so represent by 0 and next subsequent years are represented below.

$\left(0,29.3\right),\left(5,33.1\right),\left(10,33.6\right),\left(15,36.4\right),\left(18,34.5\right)$.

The slope m of a line passing through two points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is expressed below.

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

Now, compute the slope of the line passing through the points $\left(0,29.3\right)\text{ and }\left(5,33.1\right)$.

$\begin{array}{c}m=\frac{33.1-29.3}{5-0}\\ =\frac{3.8}{5}\\ =0.76\end{array}$

Therefore, slope of the line passing through the points $\left(0,29.3\right)\text{ and }\left(5,33.1\right)$ s $m=0.76$ .

Step-3 –Express the equation in point-slope form

The equation in point-slope form is expressed as $y-y_1=m\left(x-x_1\right)$.

Where m is the slope and $\left(x_1,y_1\right)$ is the point through which the line passes.

Now, compute the equation of line with slope $m=0.76$ and passing through the point $\left(0,29.3\right)$.

$\begin{array}{c}y-29.3=0.76\left(x-0\right)\\ y-29.3=0.76x\\ y=0.76x+29.3\end{array}$

Recall that equation of line in slope intercept form is expressed as $y=mx+c$

Now, the equation is in the form $y=mx+c$. Here slope m of the line is $0.76$ and intercept of y-axis c is $29.3$.

Hence, the prediction equation is $y=0.76x+29.3$.