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Sagot :
To determine the number of hats [tex]\(x\)[/tex] and T-shirts [tex]\(y\)[/tex] Pablo can buy within his constraints, we need to solve the given system of linear equations:
[tex]\[ \begin{cases} 5x + 10y = 400 \\ x + y = 50 \end{cases} \][/tex]
Let's solve it step by step:
1. First, we'll simplify the second equation to solve for one of the variables. Let's solve for [tex]\(y\)[/tex]:
[tex]\[ x + y = 50 \implies y = 50 - x \][/tex]
2. Next, we'll substitute [tex]\(y = 50 - x\)[/tex] into the first equation to eliminate [tex]\(y\)[/tex]:
[tex]\[ 5x + 10(50 - x) = 400 \][/tex]
3. Distribute the 10:
[tex]\[ 5x + 500 - 10x = 400 \][/tex]
4. Combine like terms:
[tex]\[ -5x + 500 = 400 \][/tex]
5. Isolate [tex]\(x\)[/tex]:
[tex]\[ -5x = 400 - 500 \][/tex]
[tex]\[ -5x = -100 \][/tex]
[tex]\[ x = \frac{-100}{-5} \][/tex]
[tex]\[ x = 20 \][/tex]
6. Now we have [tex]\(x = 20\)[/tex]. Substitute this back into [tex]\(y = 50 - x\)[/tex] to find [tex]\(y\)[/tex]:
[tex]\[ y = 50 - 20 \][/tex]
[tex]\[ y = 30 \][/tex]
Therefore, the solution to the system is [tex]\(x = 20\)[/tex] and [tex]\(y = 30\)[/tex]. The correct ordered pair that satisfies both equations is:
[tex]\[ (20, 30) \][/tex]
So, the correct answer is:
[tex]\[ \boxed{(20, 30)} \][/tex]
[tex]\[ \begin{cases} 5x + 10y = 400 \\ x + y = 50 \end{cases} \][/tex]
Let's solve it step by step:
1. First, we'll simplify the second equation to solve for one of the variables. Let's solve for [tex]\(y\)[/tex]:
[tex]\[ x + y = 50 \implies y = 50 - x \][/tex]
2. Next, we'll substitute [tex]\(y = 50 - x\)[/tex] into the first equation to eliminate [tex]\(y\)[/tex]:
[tex]\[ 5x + 10(50 - x) = 400 \][/tex]
3. Distribute the 10:
[tex]\[ 5x + 500 - 10x = 400 \][/tex]
4. Combine like terms:
[tex]\[ -5x + 500 = 400 \][/tex]
5. Isolate [tex]\(x\)[/tex]:
[tex]\[ -5x = 400 - 500 \][/tex]
[tex]\[ -5x = -100 \][/tex]
[tex]\[ x = \frac{-100}{-5} \][/tex]
[tex]\[ x = 20 \][/tex]
6. Now we have [tex]\(x = 20\)[/tex]. Substitute this back into [tex]\(y = 50 - x\)[/tex] to find [tex]\(y\)[/tex]:
[tex]\[ y = 50 - 20 \][/tex]
[tex]\[ y = 30 \][/tex]
Therefore, the solution to the system is [tex]\(x = 20\)[/tex] and [tex]\(y = 30\)[/tex]. The correct ordered pair that satisfies both equations is:
[tex]\[ (20, 30) \][/tex]
So, the correct answer is:
[tex]\[ \boxed{(20, 30)} \][/tex]
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