To solve the problem, we will set up a system of equations based on the information provided.
Let:
- \( x \) = number of nickels
- \( y \) = number of dimes
From the information given, we can create the following equations:
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Value Equation: The total value of the nickels and dimes is $1.20.
- The value of nickels is \( 0.05x \) (since each nickel is worth $0.05).
- The value of dimes is \( 0.10y \) (since each dime is worth $0.10).
- Thus, the equation for the total value can be written as: \[ 0.05x + 0.10y = 1.20 \] Multiplying the entire equation by 100 to eliminate decimals gives: \[ 5x + 10y = 120 \] or simplifying: \[ x + 2y = 24 \quad \text{(1)} \]
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Nickels & Dimes Relationship: Joshua has 6 more nickels than dimes. This can be expressed as: \[ x = y + 6 \quad \text{(2)} \]
Now we have a system of two equations:
- \( x + 2y = 24 \)
- \( x - y = 6 \)
Now, we can solve this system of equations graphically or algebraically. For graphical solving, we need to rearrange both equations into slope-intercept form (i.e., \( y = mx + c \)):
From equation (2): \[ x - y = 6 \implies y = x - 6 \quad \text{(3)} \]
From equation (1): \[ x + 2y = 24 \implies 2y = 24 - x \implies y = -\frac{1}{2}x + 12 \quad \text{(4)} \]
Now we can arrange equations (3) and (4) for graphing:
- Graph of Equation (3): \( y = x - 6 \) (a straight line with slope 1 and y-intercept -6).
- Graph of Equation (4): \( y = -\frac{1}{2}x + 12 \) (a straight line with slope -1/2 and y-intercept 12).
Finding the intersection:
To find the intersection point algebraically:
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Substitute equation (3) into equation (1): \[ x + 2(x - 6) = 24 \] Simplifying gives: \[ x + 2x - 12 = 24 \] \[ 3x - 12 = 24 \] \[ 3x = 36 \] \[ x = 12 \]
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Substitute \( x = 12 \) back into equation (2) to find \( y \): \[ 12 = y + 6 \implies y = 6 \]
Conclusion:
Joshua has 12 nickels and 6 dimes. In summary:
- Number of nickels, \( x = 12 \)
- Number of dimes, \( y = 6 \)
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