To determine the equation that represents the relationship between the number of bus rides, [tex]\(x\)[/tex], and the amount of bus money Shawn will have left, [tex]\(y\)[/tex], we need to follow a step-by-step analysis.
1. Identify the data given:
- Number of rides [tex]\(x = 0\)[/tex], money left [tex]\(y = \$50\)[/tex]
- Number of rides [tex]\(x = 9\)[/tex], money left [tex]\(y = \$41\)[/tex]
- Number of rides [tex]\(x = 18\)[/tex], money left [tex]\(y = \$32\)[/tex]
2. Calculate the rate at which money is being spent per ride:
- From [tex]\(x = 0\)[/tex] (rides) to [tex]\(x = 9\)[/tex] (rides), the money decreases from [tex]\( \$50 \)[/tex] to [tex]\( \$41 \)[/tex].
The rate (slope, [tex]\(m\)[/tex]) can be found using:
[tex]\[
m = \frac{\Delta y}{\Delta x} = \frac{41 - 50}{9 - 0} = \frac{-9}{9} = -1
\][/tex]
3. Determine the linear equation in slope-intercept form [tex]\( y = mx + c \)[/tex]:
- We know [tex]\(m = -1\)[/tex] (the rate found above).
- We use the point [tex]\((0, 50)\)[/tex] to find the y-intercept [tex]\(c\)[/tex].
- Since at [tex]\(x = 0\)[/tex], [tex]\(y = 50\)[/tex]:
[tex]\[
c = 50
\][/tex]
Therefore, the equation in slope-intercept form is:
[tex]\[
y = -1x + 50
\][/tex]
4. Convert to standard form [tex]\( Ax + By = C \)[/tex]:
- Our equation [tex]\( y = -x + 50 \)[/tex] can be rearranged to:
[tex]\[
x + y = 50
\][/tex]
Therefore, the equation that represents the relationship between the number of bus rides [tex]\(x\)[/tex] and the amount of bus money Shawn will have left [tex]\(y\)[/tex] in standard form is:
[tex]\[
x + y = 50
\][/tex]
