Question
Use left-to-right elimination to solve the system of equations.
left brace Start 3 By 1 Matrix 1st Row 1st Column 2nd Row 1st Column 3rd Row 1st Column EndMatrix
3 x plus 6 y minus 15 z
equals
12
4 x plus 7 y plus z
equals
52
x plus 6 y minus z
equals
36
Question content area bottom
Part 1
Select the correct choice below and, if necessary, fill in the answer boxes to complete the choice.
A.
The solution of the given system is xequals
enter your response here, yequals
enter your response here, and zequals
enter your response here.
(Simplify your answers. Type integers or simplified fractions.)
B.
The system has infinitely many solutions. If z is any real number, then xequals
enter your response here and yequals
enter your response here.
(Type expressions using z as the variable. Use integers or fractions for any numbers in the expression.)
C.
The system is inconsistent and has no solution.
left brace Start 3 By 1 Matrix 1st Row 1st Column 2nd Row 1st Column 3rd Row 1st Column EndMatrix
3 x plus 6 y minus 15 z
equals
12
4 x plus 7 y plus z
equals
52
x plus 6 y minus z
equals
36
Question content area bottom
Part 1
Select the correct choice below and, if necessary, fill in the answer boxes to complete the choice.
A.
The solution of the given system is xequals
enter your response here, yequals
enter your response here, and zequals
enter your response here.
(Simplify your answers. Type integers or simplified fractions.)
B.
The system has infinitely many solutions. If z is any real number, then xequals
enter your response here and yequals
enter your response here.
(Type expressions using z as the variable. Use integers or fractions for any numbers in the expression.)
C.
The system is inconsistent and has no solution.
Answers
Answered by
GPT-4o mini
To solve the given system of equations using left-to-right elimination, we will rewrite the equations and eliminate variables step-by-step. The system is as follows:
1. \( 3x + 6y - 15z = 12 \) (Equation 1)
2. \( 4x + 7y + z = 52 \) (Equation 2)
3. \( x + 6y - z = 36 \) (Equation 3)
**Step 1: Eliminate \( z \) from Equations 2 and 3.**
From Equation 3, we can express \( z \) in terms of \( x \) and \( y \):
\[
z = x + 6y - 36
\]
Now, substitute this expression for \( z \) into Equations 1 and 2.
**Substituting into Equation 1:**
\[
3x + 6y - 15(x + 6y - 36) = 12
\]
Expanding this:
\[
3x + 6y - 15x - 90y + 540 = 12
\]
Combining like terms:
\[
-12x - 84y + 540 = 12
\]
Subtracting 540 from both sides:
\[
-12x - 84y = -528
\]
Dividing by -12:
\[
x + 7y = 44 \quad \text{(Equation 4)}
\]
**Substituting into Equation 2:**
\[
4x + 7y + (x + 6y - 36) = 52
\]
This simplifies to:
\[
4x + 7y + x + 6y - 36 = 52
\]
Combining like terms:
\[
5x + 13y - 36 = 52
\]
Adding 36 to both sides:
\[
5x + 13y = 88 \quad \text{(Equation 5)}
\]
**Step 2: Solve the new system of equations (Equations 4 and 5).**
From Equation 4:
\[
x + 7y = 44 \quad \text{(Equation 4)}
\]
From Equation 5:
\[
5x + 13y = 88 \quad \text{(Equation 5)}
\]
**Now solve this system:**
1. From Equation 4, express \( x \):
\[
x = 44 - 7y \quad \text{(Equation 6)}
\]
2. Substitute \( x \) from Equation 6 into Equation 5:
\[
5(44 - 7y) + 13y = 88
\]
Expanding:
\[
220 - 35y + 13y = 88
\]
Combining like terms:
\[
220 - 22y = 88
\]
Subtracting 220 from both sides:
\[
-22y = -132
\]
Dividing by -22:
\[
y = 6
\]
**Step 3: Substitute \( y \) back to find \( x \) and \( z \).**
Substituting \( y = 6 \) back into Equation 6:
\[
x = 44 - 7(6)
\]
\[
x = 44 - 42 = 2
\]
Now substitute \( x \) and \( y \) back into the expression for \( z \):
\[
z = x + 6y - 36
\]
\[
z = 2 + 6(6) - 36
\]
\[
z = 2 + 36 - 36
\]
\[
z = 2
\]
**Final Solution:**
Thus, the solution to the given system of equations is:
- \( x = 2 \)
- \( y = 6 \)
- \( z = 2 \)
**So the final answer is:**
A) The solution of the given system is \( x = 2 \), \( y = 6 \), and \( z = 2 \).
1. \( 3x + 6y - 15z = 12 \) (Equation 1)
2. \( 4x + 7y + z = 52 \) (Equation 2)
3. \( x + 6y - z = 36 \) (Equation 3)
**Step 1: Eliminate \( z \) from Equations 2 and 3.**
From Equation 3, we can express \( z \) in terms of \( x \) and \( y \):
\[
z = x + 6y - 36
\]
Now, substitute this expression for \( z \) into Equations 1 and 2.
**Substituting into Equation 1:**
\[
3x + 6y - 15(x + 6y - 36) = 12
\]
Expanding this:
\[
3x + 6y - 15x - 90y + 540 = 12
\]
Combining like terms:
\[
-12x - 84y + 540 = 12
\]
Subtracting 540 from both sides:
\[
-12x - 84y = -528
\]
Dividing by -12:
\[
x + 7y = 44 \quad \text{(Equation 4)}
\]
**Substituting into Equation 2:**
\[
4x + 7y + (x + 6y - 36) = 52
\]
This simplifies to:
\[
4x + 7y + x + 6y - 36 = 52
\]
Combining like terms:
\[
5x + 13y - 36 = 52
\]
Adding 36 to both sides:
\[
5x + 13y = 88 \quad \text{(Equation 5)}
\]
**Step 2: Solve the new system of equations (Equations 4 and 5).**
From Equation 4:
\[
x + 7y = 44 \quad \text{(Equation 4)}
\]
From Equation 5:
\[
5x + 13y = 88 \quad \text{(Equation 5)}
\]
**Now solve this system:**
1. From Equation 4, express \( x \):
\[
x = 44 - 7y \quad \text{(Equation 6)}
\]
2. Substitute \( x \) from Equation 6 into Equation 5:
\[
5(44 - 7y) + 13y = 88
\]
Expanding:
\[
220 - 35y + 13y = 88
\]
Combining like terms:
\[
220 - 22y = 88
\]
Subtracting 220 from both sides:
\[
-22y = -132
\]
Dividing by -22:
\[
y = 6
\]
**Step 3: Substitute \( y \) back to find \( x \) and \( z \).**
Substituting \( y = 6 \) back into Equation 6:
\[
x = 44 - 7(6)
\]
\[
x = 44 - 42 = 2
\]
Now substitute \( x \) and \( y \) back into the expression for \( z \):
\[
z = x + 6y - 36
\]
\[
z = 2 + 6(6) - 36
\]
\[
z = 2 + 36 - 36
\]
\[
z = 2
\]
**Final Solution:**
Thus, the solution to the given system of equations is:
- \( x = 2 \)
- \( y = 6 \)
- \( z = 2 \)
**So the final answer is:**
A) The solution of the given system is \( x = 2 \), \( y = 6 \), and \( z = 2 \).