Question
5. Solve this system of equations. \( \left\{\begin{array}{l}3 x+4 y=36 \\ y=-\frac{1}{2} x+8\end{array}\right. \)
Ask by Maxwell Mullins. in the United States
Jan 22,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
La solución del sistema es \( x = 4 \) y \( y = 6 \).
Solution
Para resolver el sistema de ecuaciones:
\[
\begin{cases}
3x + 4y = 36 \\
y = -\frac{1}{2}x + 8
\end{cases}
\]
**Paso 1: Sustituir la segunda ecuación en la primera**
Sustituyamos \( y \) de la segunda ecuación en la primera ecuación:
\[
3x + 4\left(-\frac{1}{2}x + 8\right) = 36
\]
**Paso 2: Simplificar la ecuación**
Distribuimos el 4 dentro del paréntesis:
\[
3x - 2x + 32 = 36
\]
Combinamos términos semejantes:
\[
x + 32 = 36
\]
**Paso 3: Resolver para \( x \)**
Restamos 32 en ambos lados:
\[
x = 36 - 32
\]
\[
x = 4
\]
**Paso 4: Encontrar el valor de \( y \)**
Sustituimos \( x = 4 \) en la segunda ecuación:
\[
y = -\frac{1}{2}(4) + 8
\]
\[
y = -2 + 8
\]
\[
y = 6
\]
**Solución del sistema:**
\[
x = 4 \quad \text{y} \quad y = 6
\]
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Simplify this solution The Deep Dive
To solve the system of equations, we can substitute the expression for \(y\) from the second equation into the first equation. Substituting \(y = -\frac{1}{2} x + 8\) into the first equation \(3x + 4y = 36\): \[ 3x + 4\left(-\frac{1}{2}x + 8\right) = 36 \] Distributing the 4: \[ 3x - 2x + 32 = 36 \] Now, combine like terms: \[ x + 32 = 36 \] Subtract 32 from both sides: \[ x = 4 \] Now, substitute \(x = 4\) back into the equation for \(y\): \[ y = -\frac{1}{2}(4) + 8 = -2 + 8 = 6 \] The solution to the system is \(x = 4\) and \(y = 6\). So, the final answer is \((4, 6)\).
