Question

Watch the video and then solve the problem given below.
Click here to watch the video.opens in a new tab
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
Question content area bottom
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 \(z = 4\).