To solve the system of equations using left-to-right elimination, we will first write the equations clearly:
- \(3x + 4y - 5z = -6\)
- \(4x + y - z = -7\)
- \(x - 3y + 3z = -5\)
Now, let's use the elimination method step by step.
Step 1: Eliminate \(z\)
First, we can eliminate \(z\) from the first two equations. To do this, we can manipulate the equations.
From the second equation \(4x + y - z = -7\), we can express \(z\) in terms of \(x\) and \(y\): \[ z = 4x + y + 7 \]
Now, substitute \(z\) into the first equation: \[ 3x + 4y - 5(4x + y + 7) = -6 \] Distributing the -5: \[ 3x + 4y - 20x - 5y - 35 = -6 \] Combine like terms: \[ -17x - y - 35 = -6 \] Add 35 to both sides: \[ -17x - y = 29 \quad \text{(Equation 4)} \]
Step 2: Eliminate \(z\) using the third equation
Now we also substitute \(z\) into the third equation: \[ x - 3y + 3(4x + y + 7) = -5 \] Distributing: \[ x - 3y + 12x + 3y + 21 = -5 \] Combine like terms: \[ 13x + 21 = -5 \] Subtract 21 from both sides: \[ 13x = -26 \] Divide by 13: \[ x = -2 \]
Step 3: Substitute \(x\) back to find \(y\)
Now substitute \(x = -2\) into Equation 4: \[ -17(-2) - y = 29 \] Calculate: \[ 34 - y = 29 \] Subtract 34 from both sides: \[ -y = -5 \] Thus: \[ y = 5 \]
Step 4: Substitute \(x\) and \(y\) back to find \(z\)
Now we need to find \(z\). We can use any of the original equations; we’ll use the second one: \[ 4(-2) + 5 - z = -7 \] Calculate: \[ -8 + 5 - z = -7 \] \[ -3 - z = -7 \] Add 3 to both sides: \[ -z = -4 \] Thus: \[ z = 4 \]
Final Solution
The solution to the system of equations is:
- \(x = -2\)
- \(y = 5\)
- \(z = 4\)
Select Option A: A. The one solution is \(x = -2\), \(y = 5\), and \(z = 4\).