Question
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Answer
\( x=-\frac{11}{23} \) and \( y=\frac{7}{23} \).
Solution
Solve the system of equations \( 3y+4x=-1;5y-x=2 \).
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
\(\left\{ \begin{array}{l}3y+4x=-1\\5y-x=2\end{array}\right.\)
- step1: Solve the equation:
\(\left\{ \begin{array}{l}3y+4x=-1\\x=-2+5y\end{array}\right.\)
- step2: Substitute the value of \(x:\)
\(3y+4\left(-2+5y\right)=-1\)
- step3: Simplify:
\(23y-8=-1\)
- step4: Move the constant to the right side:
\(23y=-1+8\)
- step5: Add the numbers:
\(23y=7\)
- step6: Divide both sides:
\(\frac{23y}{23}=\frac{7}{23}\)
- step7: Divide the numbers:
\(y=\frac{7}{23}\)
- step8: Substitute the value of \(y:\)
\(x=-2+5\times \frac{7}{23}\)
- step9: Calculate:
\(x=-\frac{11}{23}\)
- step10: Calculate:
\(\left\{ \begin{array}{l}x=-\frac{11}{23}\\y=\frac{7}{23}\end{array}\right.\)
- step11: Check the solution:
\(\left\{ \begin{array}{l}x=-\frac{11}{23}\\y=\frac{7}{23}\end{array}\right.\)
- step12: Rewrite:
\(\left(x,y\right) = \left(-\frac{11}{23},\frac{7}{23}\right)\)
The solution to the system of equations \(3y+4x=-1\) and \(5y-x=2\) is \(x=-\frac{11}{23}\) and \(y=\frac{7}{23}\).
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Simplify this solution The Deep Dive
To solve this system of equations, you can use either substitution or elimination. For instance, if we solve for \( x \) in the second equation \( 5y - x = 2 \), we get \( x = 5y - 2 \). Next, substitute this expression for \( x \) into the first equation: \[ 3y + 4(5y - 2) = -1 \] Expanding and simplifying will lead to a single equation in \( y \). Once you find \( y \), plug it back into the expression for \( x \) to get the corresponding value. Now, a common mistake is to forget to distribute correctly or to mismanage the signs during simplifications. Double-check your work at each step to avoid those pesky errors!
