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Translate to a system of equations and solve:
A science center sold 1,363 tickets on a busy weekend. The receipts totaled $12,146. How many $12 adult tickets and how many $7 child tickets were sold?
There were 521 adult tickets sold and 842 children tickets sold.
In [link] we’ll solve a coin problem. Now that we know how to work with systems of two variables, naming the variables in the ‘number’ column will be easy.
Translate to a system of equations and solve:
Priam has a collection of nickels and quarters, with a total value of $7.30. The number of nickels is six less than three times the number of quarters. How many nickels and how many quarters does he have?
| Step 1. Read the problem. | We will create a table to organize the information. |
| Step 2. Identify what we are looking for. | We are looking for the number of nickels
and the number of quarters. |
| Step 3. Name what we are looking for. | Let
the number of nickels.
the number of quarters |
| A table will help us organize the data.
We have two types of coins, nickels and quarters. |
Write n and q for the number of each type of coin. |
| Fill in the Value column with the value of each
type of coin. |
The value of each nickel is $0.05.
The value of each quarter is $0.25. |
| The number times the value gives the total
value, so, the total value of the nickels is n (0.05) = 0.05 n and the total value of quarters is q (0.25) = 0.25 q . Altogether the total value of the coins is $7.30. |
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| Step 4. Translate into a system of equations. | |
| The Total value column gives one equation. |
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| We also know the number of nickels is six less
than three times the number of quarters. Translate to get the second equation. |
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| Now we have the system to solve. |
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Step 5. Solve the system of equations
We will use the substitution method. Substitute n = 3 q − 6 into the first equation. Simplify and solve for q . |
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| To find the number of nickels, substitute
q = 19 into the second equation. |
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Step 6. Check the answer in the problem.
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| Step 7. Answer the question. | Priam has 19 quarters and 51 nickels. |
Translate to a system of equations and solve:
Matilda has a handful of quarters and dimes, with a total value of $8.55. The number of quarters is 3 more than twice the number of dimes. How many dimes and how many quarters does she have?
Matilda has 13 dimes and 29 quarters.
Translate to a system of equations and solve:
Juan has a pocketful of nickels and dimes. The total value of the coins is $8.10. The number of dimes is 9 less than twice the number of nickels. How many nickels and how many dimes does Juan have?
Juan has 36 nickels and 63 dimes.
Some mixture applications involve combining foods or drinks. Example situations might include combining raisins and nuts to make a trail mix or using two types of coffee beans to make a blend.
Translate to a system of equations and solve:
Carson wants to make 20 pounds of trail mix using nuts and chocolate chips. His budget requires that the trail mix costs him $7.60 per pound. Nuts cost $9.00 per pound and chocolate chips cost $2.00 per pound. How many pounds of nuts and how many pounds of chocolate chips should he use?
| Step 1. Read the problem. | We will create a table to organize the information. |
| Step 2. Identify what we are looking for. | We are looking for the number of pounds of nuts
and the number of pounds of chocolate chips. |
| Step 3. Name what we are looking for. | Let
the number of pound of nuts.
the number of pounds of chips |
| Carson will mix nuts and chocolate chips
to get trail mix. Write in n and c for the number of pounds of nuts and chocolate chips. There will be 20 pounds of trail mix. Put the price per pound of each item in the Value column. Fill in the last column using |
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| Number · Value = Total Value | |
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Step 4. Translate into a system of equations.
We get the equations from the Number and Total Value columns. |
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Step 5. Solve the system of equations
We will use elimination to solve the system. |
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| Multiply the first equation by −2 to eliminate c . |
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| Simplify and add. Solve for n . |
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| To find the number of pounds of
chocolate chips, substitute n = 16 into the first equation, then solve for c . |
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Step 6. Check the answer in the problem.
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| Step 7. Answer the question. | Carson should mix 16 pounds of nuts with
4 pounds of chocolate chips to create the trail mix. |
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