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A man bought 42 stamps, some costing 13 cents and some costing 18 cents. How many of each kind did he buy if the total cost was \$6.66?

If [tex]\( x \)[/tex] represents the number of 13 cent stamps and [tex]\( y \)[/tex] represents the number of 18 cent stamps, which system of equations represents the problem?

A. [tex]\( 0.13x + 0.18y = 42 \)[/tex] and [tex]\( x + y = 6.66 \)[/tex]

B. [tex]\( x + y = 6.66 \)[/tex] and [tex]\( 0.13x = 0.18y \)[/tex]

C. [tex]\( x + y = 42 \)[/tex] and [tex]\( 0.13x + 0.18y = 6.66 \)[/tex]


Sagot :

Let's solve the problem step-by-step:

1. Identify the variables:
Let:
- [tex]\( x \)[/tex] be the number of 13 cent stamps.
- [tex]\( y \)[/tex] be the number of 18 cent stamps.

2. Set up the equations:
- The total number of stamps is 42:
[tex]\[ x + y = 42 \][/tex]
- The total cost of the stamps is [tex]$6.66. Convert dollars to cents since the stamp values are in cents. Hence, $[/tex]6.66 is equivalent to 666 cents:
[tex]\[ 0.13x + 0.18y = 6.66 \][/tex]

3. System of equations:
The system representing the problem is:
[tex]\[ \begin{cases} x + y = 42 \\ 0.13x + 0.18y = 6.66 \end{cases} \][/tex]

4. Solve the system:
We have two equations:
[tex]\[ \begin{cases} x + y = 42 \\ 0.13x + 0.18y = 6.66 \end{cases} \][/tex]

Let's solve these equations to find [tex]\( x \)[/tex] and [tex]\( y \)[/tex].

5. Isolate one variable:
From the first equation, isolate [tex]\( y \)[/tex]:
[tex]\[ y = 42 - x \][/tex]

6. Substitute into the second equation:
Substitute [tex]\( y \)[/tex] into the second equation:
[tex]\[ 0.13x + 0.18(42 - x) = 6.66 \][/tex]

7. Simplify the second equation:
[tex]\[ 0.13x + 0.18 \cdot 42 - 0.18x = 6.66 \][/tex]
[tex]\[ 0.13x + 7.56 - 0.18x = 6.66 \][/tex]
[tex]\[ -0.05x + 7.56 = 6.66 \][/tex]
[tex]\[ -0.05x = 6.66 - 7.56 \][/tex]
[tex]\[ -0.05x = -0.90 \][/tex]
[tex]\[ x = \frac{-0.90}{-0.05} \][/tex]
[tex]\[ x = 18 \][/tex]

8. Find [tex]\( y \)[/tex]:
Substitute [tex]\( x = 18 \)[/tex] back into the equation [tex]\( y = 42 - x \)[/tex]:
[tex]\[ y = 42 - 18 \][/tex]
[tex]\[ y = 24 \][/tex]

So, the man bought:
- 18 of the 13 cent stamps.
- 24 of the 18 cent stamps.

Thus, the system representing the problem is:
[tex]\[ \begin{cases} x + y = 42 \\ 0.13x + 0.18y = 6.66 \end{cases}. \][/tex]
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