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Use​ left-to-right elimination to solve the system of equations.
left brace Start 2 By 1 Matrix 1st Row 1st Column 2nd Row 1st Column EndMatrix

3 x plus
4 y minus
5 z
equalsnegative 6
4 x plus
y minus
z
equalsnegative 7
x minus
3 y plus
3 z
equalsnegative 5
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Part 1
Select the correct choice below and fill in any answer boxes within your choice.
A.
The one solution is xequals
  
enter your response here​, yequals
  
enter your response here​, and zequals
  
enter your response here. ​(Simplify your​ answers.)
B.
There are infinitely many solutions. If z is allowed to be any real​ number, then xequals
  
enter your response here and yequals
  
enter your response here.
​(Type expressions using z as the​ variable.)
C.

Answer 1

To solve the system of equations using left-to-right elimination, we will first write the equations clearly:

1. \(3x + 4y - 5z = -6\)
2. \(4x + y - z = -7\)
3. \(x - 3y + 3z = -5\)

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 \(z = 4\).