Problem 8 - Beginning and Intermediate Algebra

Question

\begin{aligned} -2 x-2 y+3 z &=2 \\ 3 x+3 y-5 z &=-3 \\ -x+y-z &=9 \end{aligned}

Step-by-Step Solution

Answer

The short version answer for the given system of linear equations is: \(x = -5\), \(y = 4\), and \(z = 0\).

Step 1: Solve the third equation for x

From the third equation, you can easily find x in terms of y and z: \begin{aligned} -x + y - z &= 9 \Rightarrow x = y - z - 9 \end{aligned}

Step 2: Substitute x into the first two equations

Replace x in the first two equations with the expression from Step 1: \((y-z-9)\): \begin{aligned} 1. & -2(y - z - 9) - 2y+3z &= 2 \\ 2. & 3(y - z - 9) + 3y-5z &= -3 \end{aligned}

Step 3: Simplify the two new equations

Distribute and simplify the two new equations: \begin{aligned} 1. &-2y +2z +18 - 2y+3z &= 2 \\ 2. & 3y -3z - 27+3y -5z &= -3 \end{aligned} Combine like terms: \begin{aligned} 1. &-4y+5z &= -16 \\ 2. & 6y -8z &= 24 \end{aligned}

Step 4: Solve the 2x2 system of equations

In this step, we will solve the 2x2 system of equations. Multiply the first equation by 2 and add the equations to eliminate y: \begin{aligned} 2(-4y+5z) + (6y-8z) &= -32 + 24 \end{aligned} Combine like terms: \begin{aligned} -2y + 2z &= -8 \end{aligned} Now, solve this equation for z: \begin{aligned} z &= y - 4 \end{aligned}

Step 5: Substitute z back into an equation to find y

Substitute the expression for z back into one of the original equations, such as the first equation: \begin{aligned} -4(y) + 5(y - 4) &= -16 \\ \end{aligned} Distribute and simplify: \begin{aligned} -4y+5y - 20 &= -16 \\ \end{aligned} Solve for y: \begin{aligned} y = 4 \end{aligned}

Step 6: Find x and z using the values of y

Substitute the value of y back into the expressions we found earlier for x and z: \begin{aligned} z &= y - 4 = 4 - 4 = 0 \\ x &= y - z - 9 = 4 - 0 - 9 = -5 \end{aligned} The solution to the system of equations is \(x = -5\), \(y = 4\), and \(z = 0\).