Question

Eric has a gift card to his favorite restaurant. Every Tuesday he has the same meal at the restaurant. This table shows the balance of money in dollars left on Eric’s gift card at different times. Time (weeks) 2 4 6 8 Balance ($) 125 100 75 50 What is the equation that represents Eric’s gift card balance, where x represents the time and y represents the balance? Enter your answers in the boxes. y−100= (x− )

Answer 1

Answer-

$y-100=-\frac{25}{2}(x-4)$

Solution-

We can generate an equation that will represent Eric’s gift card balance, by taking two points from the table and applying two point formula to get the linear relationship between card balance and time.

Taking the time on abscissa and card balance on ordinate, the two points would be (2, 125), (4, 100)

$\Rightarrow \frac{y-y_1}{y_2-y_1}=\frac{x-x_1}{x_2-x_1}$

$\Rightarrow \frac{y-125}{100-125}=\frac{x-2}{4-2}$

$\Rightarrow \frac{y-125}{-25}=\frac{x-2}{2}$

$\Rightarrow -\frac{y}{25}+5=\frac{x}{2}-1$

$\Rightarrow \frac{x}{2}+\frac{y}{25}-6=0$

$\Rightarrow \frac{25x+2y-300}{50}=0$

$\Rightarrow 25x+2y-300=0$

$\Rightarrow y=\frac{300-25x}{2}$

$\Rightarrow y=150-\frac{25}{2}x$

This is the equation for the Eric’s gift card balance.

Comparing this equation with general slope-intercept equation of the line, we get

$Slope=-\frac{25}{2} \ and\ y-intercept=150$

Applying point-slope formula taking point as (4, 100), we get

$\Rightarrow y-y_1=m(x-x_1)$

$\Rightarrow y-100=-\frac{25}{2}(x-4)$

This is the desired equation.