Question
Solve the simultaneous equation \( 2 x+5 y=9 \) \( 2 x+3 y=7 \)
Ask by Pearson Wilson. in South Africa
Mar 12,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( x=2 \) and \( y=1 \).
Solution
Solve the system of equations \( 2x+5y=9;2x+3y=7 \).
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
\(\left\{ \begin{array}{l}2x+5y=9\\2x+3y=7\end{array}\right.\)
- step1: Solve the equation:
\(\left\{ \begin{array}{l}x=\frac{9-5y}{2}\\2x+3y=7\end{array}\right.\)
- step2: Substitute the value of \(x:\)
\(2\times \frac{9-5y}{2}+3y=7\)
- step3: Simplify:
\(9-2y=7\)
- step4: Move the constant to the right side:
\(-2y=7-9\)
- step5: Subtract the numbers:
\(-2y=-2\)
- step6: Change the signs:
\(2y=2\)
- step7: Divide both sides:
\(\frac{2y}{2}=\frac{2}{2}\)
- step8: Divide the numbers:
\(y=1\)
- step9: Substitute the value of \(y:\)
\(x=\frac{9-5\times 1}{2}\)
- step10: Substitute back:
\(x=\frac{9-5}{2}\)
- step11: Calculate:
\(x=2\)
- step12: Calculate:
\(\left\{ \begin{array}{l}x=2\\y=1\end{array}\right.\)
- step13: Check the solution:
\(\left\{ \begin{array}{l}x=2\\y=1\end{array}\right.\)
- step14: Rewrite:
\(\left(x,y\right) = \left(2,1\right)\)
The solution to the simultaneous equations \(2x+5y=9\) and \(2x+3y=7\) is \(x=2\) and \(y=1\).
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Simplify this solution Bonus Knowledge
To solve the simultaneous equations, we can start by eliminating one variable. Let's subtract the second equation from the first: \[ (2x + 5y) - (2x + 3y) = 9 - 7 \] This simplifies to: \[ 2y = 2 \] Dividing both sides by 2 gives us: \[ y = 1 \] Now that we have \(y\), we can substitute it back into one of the original equations. Using the first equation: \[ 2x + 5(1) = 9 \] This simplifies to: \[ 2x + 5 = 9 \] Subtracting 5 from both sides gives: \[ 2x = 4 \] Dividing by 2, we find: \[ x = 2 \] Therefore, the solution to the simultaneous equations is \(x = 2\) and \(y = 1\).
