To solve the problem, we need to set up a system of equations based on the information provided.
- Let \( x \) be the number of nickels.
- Let \( y \) be the number of dimes.
- Each nickel is worth $0.05, and each dime is worth $0.10.
- The total value of the nickels and dimes is $2, which we can express as: \[ 0.05x + 0.10y = 2 \]
- 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:
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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)
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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.