2x- y +3z＝11
2x+3y - z =-9
x + 3y +2 = -6

The given system of linear equations is:

1. \( 2x - y + 3z = 11 \)
2. \( 2x + 3y - z = -9 \)
3. \( x + 3y + 2 = -6 \)

### Step 1: Simplify the third equation
Let's simplify the third equation:
\[
x + 3y + 2 = -6
\]
Subtract 2 from both sides:
\[
x + 3y = -8 \quad \text{(Equation 3 simplified)}
\]

So now we have the system:
1. \( 2x - y + 3z = 11 \)
2. \( 2x + 3y - z = -9 \)
3. \( x + 3y = -8 \)

### Step 2: Use substitution or elimination
Let's solve this system using substitution or elimination. First, we can solve for \(x\) from the third equation:
\[
x = -8 - 3y
\]
Now, substitute this value of \(x\) into the first two equations.

#### Substituting into Equation 1:
\[
2(-8 - 3y) - y + 3z = 11
\]
Simplify:
\[
-16 - 6y - y + 3z = 11
\]
\[
-16 - 7y + 3z = 11
\]
Add 16 to both sides:
\[
-7y + 3z = 27 \quad \text{(Equation A)}
\]

#### Substituting into Equation 2:
\[
2(-8 - 3y) + 3y - z = -9
\]
Simplify:
\[
-16 - 6y + 3y - z = -9
\]
\[
-16 - 3y - z = -9
\]
Add 16 to both sides:
\[
-3y - z = 7 \quad \text{(Equation B)}
\]

### Step 3: Solve the system of two equations
Now we have the system:
1. \( -7y + 3z = 27 \) (Equation A)
2. \( -3y - z = 7 \) (Equation B)

Let's solve for \(z\) from Equation B:
\[
z = -3y - 7
\]
Substitute this into Equation A:
\[
-7y + 3(-3y - 7) = 27
\]
Simplify:
\[
-7y - 9y - 21 = 27
\]
\[
-16y - 21 = 27
\]
Add 21 to both sides:
\[
-16y = 48
\]
Solve for \(y\):
\[
y = -3
\]

### Step 4: Solve for \(z\)
Substitute \(y = -3\) into \(z = -3y - 7\):
\[
z = -3(-3) - 7 = 9 - 7 = 2
\]

### Step 5: Solve for \(x\)
Substitute \(y = -3\) into \(x = -8 - 3y\):
\[
x = -8 - 3(-3) = -8 + 9 = 1
\]

### Solution
The solution to the system is:
\[
x = 1, \quad y = -3, \quad z = 2
\]

Would you like more details or have any questions? 

Here are 5 related questions to expand your knowledge:
1. How do you solve a system of equations using the substitution method?
2. What is the elimination method in solving systems of equations?
3. How do you determine if a system of equations has a unique solution, infinite solutions, or no solution?
4. How can matrix methods (e.g., Gaussian elimination) be applied to solve systems of linear equations?
5. What role does linear algebra play in solving systems of equations with more variables?

**Tip:** Always double-check your work by substituting the values back into the original equations to ensure they hold true.

2x - y + 3z = 11
2x + 3y - z = -9
x + 3y + 2 = -6

Explore how to solve the system of linear equations 2x - y + 3z = 11, 2x + 3y - z = -9, and x + 3y + 2 = -6 using substitution and elimination methods. This problem is ideal for students in Grades 9-11 and covers key algebraic concepts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation