Sure, let's go through the process of creating the equation that represents the value of the money John has in terms of [tex]\( x \)[/tex], the number of dimes John has.
1. Identify the value of each type of coin:
- The value of one dime is [tex]$0.10.
- The value of one nickel is $[/tex]0.05.
2. Express the number of nickels in terms of the number of dimes:
- According to the problem, John has 4 more nickels than dimes.
- If [tex]\( x \)[/tex] represents the number of dimes, then the number of nickels can be expressed as [tex]\( x + 4 \)[/tex].
3. Calculate the total value of the dimes:
- The value of [tex]\( x \)[/tex] dimes is [tex]\( 0.10x \)[/tex] dollars.
4. Calculate the total value of the nickels:
- The number of nickels is [tex]\( x + 4 \)[/tex].
- Thus, the value of the nickels is [tex]\( 0.05(x + 4) \)[/tex] dollars.
5. Combine the value of the dimes and nickels to get the total value:
- The total value, [tex]\( y \)[/tex], is the sum of the value of the dimes and the value of the nickels.
- Therefore, the equation is:
[tex]\[
y = 0.10x + 0.05(x + 4)
\][/tex]
6. Simplify the equation if needed:
- Although simplification is optional here, it can make the equation look cleaner or easier to interpret.
Now, let's discuss the y-intercept of the equation.
The y-intercept is the value of [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex].
1. Determine the y-intercept:
- Substitute [tex]\( x = 0 \)[/tex] into the equation [tex]\( y = 0.10x + 0.05(x + 4) \)[/tex]:
[tex]\[
y = 0.10(0) + 0.05(0 + 4)
\][/tex]
- Simplify it:
[tex]\[
y = 0 + 0.05 \cdot 4
\][/tex]
- Calculate [tex]\( 0.05 \times 4 \)[/tex]:
[tex]\[
y = 0.20
\][/tex]
2. Interpret the y-intercept:
- The y-intercept, in this case, represents the total value of the nickels when the number of dimes ([tex]\( x \)[/tex]) is zero.
- So, if John has zero dimes, he still has the value of four nickels, which is $0.20.
In summary:
- The equation representing the total value of John's money in terms of [tex]\( x \)[/tex], the number of dimes, is:
[tex]\[
y = 0.10x + 0.05(x + 4)
\][/tex]
- The y-intercept of this equation is 0.20, which represents the value of the nickels when John has no dimes.
