Question

Gabriel has $2 worth of nickels and dimes. He has 2 more dimes than nickels. Graphically solve a system of equations in order to determine the number of nickels, x, commax, and the number of dimes, y, commay, that Gabriel has.

Answer 1

To solve the problem, we need to set up a system of equations based on the information provided.

1. Let \( x \) be the number of nickels.
2. Let \( y \) be the number of dimes.
3. Each nickel is worth $0.05, and each dime is worth $0.10.
4. The total value of the nickels and dimes is $2, which we can express as:
\[
0.05x + 0.10y = 2
\]
5. Gabriel has 2 more dimes than nickels, which we can express as:
\[
y = x + 2
\]

Now we have the following system of equations:

\[
\begin{align*}
0.05x + 0.10y &= 2 \quad \text{(1)}\\
y &= x + 2 \quad \text{(2)}
\end{align*}
\]

To solve this system, we can substitute equation (2) into equation (1).

Substituting \( y \) in equation (1):
\[
0.05x + 0.10(x + 2) = 2
\]
Distributing \( 0.10 \):
\[
0.05x + 0.10x + 0.20 = 2
\]
Combining like terms:
\[
0.15x + 0.20 = 2
\]
Subtracting \( 0.20 \) from both sides:
\[
0.15x = 1.80
\]
Dividing both sides by \( 0.15 \):
\[
x = \frac{1.80}{0.15} = 12
\]

Now we know Gabriel has 12 nickels. We can find the number of dimes using equation (2):
\[
y = x + 2 = 12 + 2 = 14
\]

So, Gabriel has:
- **12 nickels** (x = 12)
- **14 dimes** (y = 14)

### Summary:
- **Number of Nickels (x):** 12
- **Number of Dimes (y):** 14

### Graphical Representation:
1. **Equation (1):** \( 0.05x + 0.10y = 2 \) can be rearranged to find \( y \):
\[
y = 20 - 0.5x
\]
(This line intercepts the y-axis at 20 and has a slope of -0.5)

2. **Equation (2):** \( y = x + 2 \) is a straight line with a slope of 1, intercepting the y-axis at 2.

Both these lines can be graphed on a coordinate plane, and the intersection point of the two lines will be at (12, 14). This corresponds to the solution we found algebraically.