To explain the intercepts in the context of the given equation [tex]\(5q + 2d = 270\)[/tex], we first need to understand what intercepts signify.
### Context
In this problem, the equation [tex]\(5q + 2d = 270\)[/tex] represents the total value (in cents) of quarters ([tex]\(q\)[/tex]) and dimes ([tex]\(d\)[/tex]) in Austin's piggy bank. Here, 'q' stands for the number of quarters, each worth 25 cents, and 'd' stands for the number of dimes, each worth 10 cents.
### Intercepts
Intercepts of the equation in this context indicate the points where either the number of quarters or the number of dimes is zero. These are helpful in understanding the extreme scenarios:
- x-intercept (quarters): The point where the number of dimes is zero.
- y-intercept (dimes): The point where the number of quarters is zero.
### Finding the Intercepts
#### Finding the y-Intercept (when the number of quarters [tex]\( q \)[/tex] is 0):
1. Set [tex]\( q = 0 \)[/tex] in the equation [tex]\( 5q + 2d = 270 \)[/tex].
2. Substitute [tex]\( q = 0 \)[/tex] into the equation:
[tex]\[
5(0) + 2d = 270
\][/tex]
3. Simplify the equation:
[tex]\[
2d = 270
\][/tex]
4. Solve for [tex]\( d \)[/tex]:
[tex]\[
d = \frac{270}{2} = 135
\][/tex]
So, the y-intercept is 135, which means if Austin has no quarters, he would have 135 dimes to make up the 270 cents.
#### Finding the x-Intercept (when the number of dimes [tex]\( d \)[/tex] is 0):
1. Set [tex]\( d = 0 \)[/tex] in the equation [tex]\( 5q + 2d = 270 \)[/tex].
2. Substitute [tex]\( d = 0 \)[/tex] into the equation:
[tex]\[
5q + 2(0) = 270
\][/tex]
3. Simplify the equation:
[tex]\[
5q = 270
\][/tex]
4. Solve for [tex]\( q \)[/tex]:
[tex]\[
q = \frac{270}{5} = 54
\][/tex]
So, the x-intercept is 54, which means if Austin has no dimes, he would have 54 quarters to make up the 270 cents.
### Interpretation
- The x-intercept (54) tells us that if Austin has 0 dimes, he would need 54 quarters to have 270 cents.
- The y-intercept (135) tells us that if Austin has 0 quarters, he would need 135 dimes to have 270 cents.
These intercepts provide valuable insights into the possible combinations of quarters and dimes that can make up a total of 270 cents in Austin's piggy bank.
