13. COINS Declan has 17 coins in his pocket. All the coins are dimes and quarters. The value of the coins is 290 cents. Define v
2 answers:
Answer:
- 9 dimes
- 8 quarters
Step-by-step explanation:
The problem requests numbers of dimes and quarters, so we can assign a variable to each of those numbers. Let x and y represent the numbers of dimes and quarters Declan has, respectively. (If we were solving this by hand, we might choose d and q to remind us what they stand for. Solving graphically, we use variables the graphing calculator recognizes.)
The problem statement tells us two relationships between these variables:
x + y = 17 . . . . . the total number of coins is 17
10x +25y = 290 . . . . . the total value of the coins is 290 cents
(Note that a dime is worth 10 cents, so the number of cents represented by x dimes is 10x cents.)
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The attached graph shows the solution to these equations. It is the point where the lines cross: (x, y) = (9, 8).
Declan has 9 dimes and 8 quarters.
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Additional comment
We prefer a graphing calculator for finding graphical solutions.
If you're doing this by hand, you can graph the equations using their x- and y-intercepts. Each of those is found by solving the equation when the other variable is zero.
These equations are in "standard form:"
ax +by = c
Then the intercepts are ...
x-intercept = c/a
y-intercept = c/b
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The y-intercept corresponding to the number of quarters is not on an integer point. (It is y = 290/25 = 11.6.) To plot the line for the second equation, you may want to choose y=10, so x=4. The point (4, 10) will be on that line, making the line easier to plot.
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