Marsh Norris
04/02/2024 · Junior High School
\( \left. \begin{array} { l } { 1 ) } \\ { 2 x + 3 y = 7 } \\ { 5 x + 2 y = 12 } \\ \hline \operatorname { Solución } ( 2,1 ) \\ \hline \end{array} \right. \)
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Step-by-step Solution
To solve the system of equations given by:
1. \( 2x + 3y = 7 \)
2. \( 5x + 2y = 12 \)
we can use either the substitution method or the elimination method. Here, I'll use the elimination method.
First, let's multiply the first equation by 5 and the second equation by 2 to align the coefficients of \(x\):
\[
5(2x + 3y) = 5(7) \implies 10x + 15y = 35 \quad \text{(Equation 3)}
\]
\[
2(5x + 2y) = 2(12) \implies 10x + 4y = 24 \quad \text{(Equation 4)}
\]
Now we have the following system:
3. \( 10x + 15y = 35 \)
4. \( 10x + 4y = 24 \)
Next, we can subtract Equation 4 from Equation 3:
\[
(10x + 15y) - (10x + 4y) = 35 - 24
\]
\[
15y - 4y = 11
\]
\[
11y = 11
\]
\[
y = 1
\]
Now that we have \(y\), we can substitute it back into one of the original equations to find \(x\). Let's use the first equation:
\[
2x + 3(1) = 7
\]
\[
2x + 3 = 7
\]
\[
2x = 7 - 3
\]
\[
2x = 4
\]
\[
x = 2
\]
Thus, the solution to the system of equations is:
\[
(x, y) = (2, 1)
\]
This matches the solution you provided: \( (2, 1) \).
Quick Answer
The solution to the system of equations is \( (x, y) = (2, 1) \).
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