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Andria and Beto both collect comic books. If Andria gave Beto 10 comic books, they would have the same number. If Beto gave Andria 12 comic books, then Andria would have twice as many as Beto. How many comic books does each one actually have?

Use variables to represent unknowns. Always define your variables to avoid confusion with your solution. Use letters that help you remember what each unknown stands for, like [tex]A[/tex] and [tex]B[/tex] here.

Let [tex]A[/tex] be the number of comic books Andria has.
Let [tex]B[/tex] be the number of comic books Beto has.

To find the values of two unknowns, you need to write two equations. (If there was only one unknown, you'd need only one equation. Three equations are needed for three unknowns.)

To write the equations, translate the sentences with numbers into algebra:
1. "If Andria gave Beto 10 comic books, they would have the same number."
[tex]\[ A - 10 = B + 10 \][/tex]
They have the same number after Andria gives away 10 and Beto gets 10.

2. "If Beto gave Andria 12 comic books, then Andria would have twice as many as Beto."
[tex]\[ A + 12 = 2(B - 12) \][/tex]
When Andria has 12 more, she will have twice as many as Beto after he gives away 12.

To solve the system of equations, solve one equation for one variable and substitute that into the other equation:
1. From [tex]\[ A - 10 = B + 10 \][/tex]
[tex]\[ A = B + 20 \][/tex] (Add 10 to both sides)

2. Substitute [tex]\[ A = B + 20 \][/tex] into [tex]\[ A + 12 = 2(B - 12) \][/tex]
[tex]\[ (B + 20) + 12 = 2(B - 12) \][/tex]

3. Simplify:
[tex]\[ B + 32 = 2B - 24 \][/tex]

4. Solve for [tex]B[/tex]:
[tex]\[ 32 + 24 = 2B - B \][/tex]
[tex]\[ 56 = B \][/tex]

Thus, Beto has 56 comic books.

To find the number of comic books Andria has, use [tex] A = B + 20 [/tex]:
[tex]\[ A = 56 + 20 \][/tex]
[tex]\[ A = 76 \][/tex]

Therefore, Andria has 76 comic books.

Sagot :

GinnyAnsweridn
Let's define our variables clearly:
- [tex]\( A \)[/tex] represents the number of comic books Andria has.
- [tex]\( B \)[/tex] represents the number of comic books Beto has.

Given the conditions in the problem, we formulate our equations as follows:

### First Condition
"If Andria gave Beto 10 comic books, they would have the same number."

This translates to:
[tex]\[ A - 10 = B + 10 \][/tex]

Solving for [tex]\( A \)[/tex]:
[tex]\[ A - B = 20 \][/tex]
[tex]\[ A = B + 20 \quad \text{(Equation 1)} \][/tex]

### Second Condition
"If Beto gave Andria 12 comic books, then Andria would have twice as many as Beto."

This translates to:
[tex]\[ A + 12 = 2(B - 12) \][/tex]

### Substitution
Using Equation 1 ([tex]\( A = B + 20 \)[/tex]) in the second equation:
[tex]\[ (B + 20) + 12 = 2(B - 12) \][/tex]

Simplifying the equation:
[tex]\[ B + 32 = 2B - 24 \][/tex]
[tex]\[ B + 32 + 24 = 2B \][/tex]
[tex]\[ B + 56 = 2B \][/tex]

Solving for [tex]\( B \)[/tex]:
[tex]\[ 56 = 2B - B \][/tex]
[tex]\[ 56 = B \][/tex]

So, Beto has:
[tex]\[ B = 56 \][/tex]

Substituting [tex]\( B = 56 \)[/tex] back into Equation 1 to find [tex]\( A \)[/tex]:
[tex]\[ A = B + 20 \][/tex]
[tex]\[ A = 56 + 20 \][/tex]
[tex]\[ A = 76 \][/tex]

So, Andria has:
[tex]\[ A = 76 \][/tex]

### Conclusion
Andria has 76 comic books, and Beto has 56 comic books.
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