Question
9) \( \begin{array}{l}-5 x+y=-2 \\ -3 x+6 y=-12\end{array} \)
Ask by Morgan Brewer. in the United States
Feb 25,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The solution is \( x = 0 \) and \( y = -2 \).
Solution
Solve the system of equations \( -5x+y=-2;-3x+6y=-12 \).
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
\(\left\{ \begin{array}{l}-5x+y=-2\\-3x+6y=-12\end{array}\right.\)
- step1: Solve the equation:
\(\left\{ \begin{array}{l}y=-2+5x\\-3x+6y=-12\end{array}\right.\)
- step2: Substitute the value of \(y:\)
\(-3x+6\left(-2+5x\right)=-12\)
- step3: Simplify:
\(27x-12=-12\)
- step4: Move the constant to the right side:
\(27x=-12+12\)
- step5: Add the numbers:
\(27x=0\)
- step6: Rewrite the expression:
\(x=0\)
- step7: Substitute the value of \(x:\)
\(y=-2+5\times 0\)
- step8: Substitute back:
\(y=-2\)
- step9: Calculate:
\(\left\{ \begin{array}{l}x=0\\y=-2\end{array}\right.\)
- step10: Check the solution:
\(\left\{ \begin{array}{l}x=0\\y=-2\end{array}\right.\)
- step11: Rewrite:
\(\left(x,y\right) = \left(0,-2\right)\)
The solution to the system of equations is \( (x, y) = (0, -2) \).
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Simplify this solution Bonus Knowledge
To solve the system of equations, we can use substitution or elimination. Let's go with elimination. First, we can rearrange the first equation: \(y = 5x - 2\). Next, we substitute this expression for y into the second equation: \[ -3x + 6(5x - 2) = -12. \] Simplifying gives: \[ -3x + 30x - 12 = -12, \] \[ 27x - 12 = -12, \] \[ 27x = 0 \Rightarrow x = 0. \] Now, substituting \(x = 0\) back into the first equation yields \(y = 5(0) - 2 = -2\). Thus, the solution to the system is \(x = 0\) and \(y = -2\). Additionally, check both equations using these values to ensure consistency. Both return valid equality, confirming \( \boxed{(0, -2)} \) as the correct answer.
