Question
12) \( \begin{aligned}-7 x+y & =-19 \\ -2 x+3 y & =-19\end{aligned} \)
Ask by Warren Bartlett. in the United States
Feb 25,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( x = 2 \) and \( y = -5 \).
Solution
Solve the system of equations \( -7x+y=-19;-2x+3y=-19 \).
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
\(\left\{ \begin{array}{l}-7x+y=-19\\-2x+3y=-19\end{array}\right.\)
- step1: Solve the equation:
\(\left\{ \begin{array}{l}y=-19+7x\\-2x+3y=-19\end{array}\right.\)
- step2: Substitute the value of \(y:\)
\(-2x+3\left(-19+7x\right)=-19\)
- step3: Simplify:
\(19x-57=-19\)
- step4: Move the constant to the right side:
\(19x=-19+57\)
- step5: Add the numbers:
\(19x=38\)
- step6: Divide both sides:
\(\frac{19x}{19}=\frac{38}{19}\)
- step7: Divide the numbers:
\(x=2\)
- step8: Substitute the value of \(x:\)
\(y=-19+7\times 2\)
- step9: Calculate:
\(y=-5\)
- step10: Calculate:
\(\left\{ \begin{array}{l}x=2\\y=-5\end{array}\right.\)
- step11: Check the solution:
\(\left\{ \begin{array}{l}x=2\\y=-5\end{array}\right.\)
- step12: Rewrite:
\(\left(x,y\right) = \left(2,-5\right)\)
The solution to the system of equations is \( x = 2 \) and \( y = -5 \).
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Simplify this solution Bonus Knowledge
To solve the system of equations, we can use the substitution or elimination method. Let's use the elimination method. First, we can align the equations: 1) \( -7x + y = -19 \) 2) \( -2x + 3y = -19 \) From equation (1), we can express \( y \) in terms of \( x \): \( y = -19 + 7x \). Now, substituting this expression for \( y \) into equation (2): \(-2x + 3(-19 + 7x) = -19\) \(-2x - 57 + 21x = -19\) \(19x - 57 = -19\) \(19x = 38\) \(x = 2\). Now substitute \( x = 2 \) back into equation (1) to find \( y \): \(-7(2) + y = -19\) \(-14 + y = -19\) \(y = -19 + 14\) \(y = -5\). Thus, the solution to the system of equations is \( x = 2 \) and \( y = -5 \).
