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Solve the simultaneous equations. Show your working. \[ \left\{\begin{array}{l}4 x-3 y=28 \\ 6 x+y=9\end{array} \begin{array}{r}4 x-3 y=28 \\ 6 x+y=9\end{array}\right. \]

Ask by Griffin Parry. in Pakistan
Jan 24,2025

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\( x = \frac{5}{2} \) and \( y = -6 \).

Solución

Solve the system of equations \( 4x-3y=28;6x+y=9 \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}4x-3y=28\\6x+y=9\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}4x-3y=28\\y=9-6x\end{array}\right.\) - step2: Substitute the value of \(y:\) \(4x-3\left(9-6x\right)=28\) - step3: Simplify: \(22x-27=28\) - step4: Move the constant to the right side: \(22x=28+27\) - step5: Add the numbers: \(22x=55\) - step6: Divide both sides: \(\frac{22x}{22}=\frac{55}{22}\) - step7: Divide the numbers: \(x=\frac{5}{2}\) - step8: Substitute the value of \(x:\) \(y=9-6\times \frac{5}{2}\) - step9: Simplify: \(y=-6\) - step10: Calculate: \(\left\{ \begin{array}{l}x=\frac{5}{2}\\y=-6\end{array}\right.\) - step11: Check the solution: \(\left\{ \begin{array}{l}x=\frac{5}{2}\\y=-6\end{array}\right.\) - step12: Rewrite: \(\left(x,y\right) = \left(\frac{5}{2},-6\right)\) The solution to the simultaneous equations is \( x = \frac{5}{2} \) and \( y = -6 \).

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To solve the simultaneous equations, we'll use the substitution or elimination method. Let's start with the equations: 1. \( 4x - 3y = 28 \) (Equation 1) 2. \( 6x + y = 9 \) (Equation 2) We can solve Equation 2 for \( y \): \[ y = 9 - 6x \] Now, we substitute this expression for \( y \) into Equation 1: \[ 4x - 3(9 - 6x) = 28 \] Expanding this we get: \[ 4x - 27 + 18x = 28 \] Combining like terms: \[ 22x - 27 = 28 \] Adding 27 to both sides gives: \[ 22x = 55 \] Dividing both sides by 22, we find: \[ x = \frac{55}{22} = 2.5 \] Now we substitute \( x \) back into the equation for \( y \): \[ y = 9 - 6(2.5) \] \[ y = 9 - 15 \] \[ y = -6 \] Thus, the solution to the simultaneous equations is: \[ (x, y) = (2.5, -6) \] To verify, we can plug \( x \) and \( y \) back into both original equations: 1. Checking Equation 1: \[ 4(2.5) - 3(-6) = 10 + 18 = 28 \] ✓ 2. Checking Equation 2: \[ 6(2.5) + (-6) = 15 - 6 = 9 \] ✓ Both equations are satisfied, confirming that the solution is correct!

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