Question
Solve the system by substitution: \[ \begin{array}{l}5 \mathrm{x}+4 \mathrm{y}=-14 \\ \mathrm{y}=-7 \mathrm{x}-15\end{array} \]
Ask by Rowe Wilson. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( x = -2 \) and \( y = -1 \).
Solution
Solve the system of equations \( 5x+4y=-14;y=-7x-15 \).
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
\(\left\{ \begin{array}{l}5x+4y=-14\\y=-7x-15\end{array}\right.\)
- step1: Substitute the value of \(y:\)
\(5x+4\left(-7x-15\right)=-14\)
- step2: Simplify:
\(-23x-60=-14\)
- step3: Move the constant to the right side:
\(-23x=-14+60\)
- step4: Add the numbers:
\(-23x=46\)
- step5: Change the signs:
\(23x=-46\)
- step6: Divide both sides:
\(\frac{23x}{23}=\frac{-46}{23}\)
- step7: Divide the numbers:
\(x=-2\)
- step8: Substitute the value of \(x:\)
\(y=-7\left(-2\right)-15\)
- step9: Simplify:
\(y=-1\)
- step10: Calculate:
\(\left\{ \begin{array}{l}x=-2\\y=-1\end{array}\right.\)
- step11: Check the solution:
\(\left\{ \begin{array}{l}x=-2\\y=-1\end{array}\right.\)
- step12: Rewrite:
\(\left(x,y\right) = \left(-2,-1\right)\)
The solution to the system of equations by substitution is \( x = -2 \) and \( y = -1 \).
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To solve the system using substitution, we can plug the expression for \(y\) from the second equation into the first equation. So, we have: 1. First equation: \(5x + 4y = -14\) 2. Second equation: \(y = -7x - 15\) Now, substitute the expression for \(y\) into the first equation: \[ 5x + 4(-7x - 15) = -14 \] Now simplify the equation: \[ 5x - 28x - 60 = -14 \] \[ -23x - 60 = -14 \] Next, add 60 to both sides: \[ -23x = 46 \] Now, divide both sides by -23: \[ x = -2 \] Now that we have \(x\), let's find \(y\) using the second equation: \[ y = -7(-2) - 15 \] \[ y = 14 - 15 \] \[ y = -1 \] Thus, the solution to the system of equations is: \[ (x, y) = (-2, -1) \]
