8x−y+7z=
−2
3, x, minus, y, plus, z, equals, minus, 9
3x−y+z=
−9
5, x, plus, y, minus, 6, z, equals, minus, 9
5x+y−6z=
−9
solve this system of equations

You provided a system of three linear equations:

\[
\begin{align*}
1) & \ 8x - y + 7z = -2 \\
2) & \ 3x - y + z = -9 \\
3) & \ 5x + y - 6z = -9
\end{align*}
\]

Let's solve this system using the substitution or elimination method.

### Step 1: Solve for \( y \) from one of the equations.

From equation \( (2) \):
\[
3x - y + z = -9
\]
Rearrange to solve for \( y \):
\[
y = 3x + z + 9
\]

### Step 2: Substitute \( y \) into the other two equations.

Substitute \( y = 3x + z + 9 \) into equation \( (1) \) and equation \( (3) \).

#### For equation \( (1) \):
\[
8x - (3x + z + 9) + 7z = -2
\]
Simplify:
\[
8x - 3x - z - 9 + 7z = -2
\]
\[
5x + 6z - 9 = -2
\]
\[
5x + 6z = 7 \quad \text{(Equation 4)}
\]

#### For equation \( (3) \):
\[
5x + (3x + z + 9) - 6z = -9
\]
Simplify:
\[
5x + 3x + z + 9 - 6z = -9
\]
\[
8x - 5z + 9 = -9
\]
\[
8x - 5z = -18 \quad \text{(Equation 5)}
\]

### Step 3: Solve the two new equations.

Now, we have a system of two equations:
\[
\begin{align*}
4) & \ 5x + 6z = 7 \\
5) & \ 8x - 5z = -18
\end{align*}
\]

Multiply equation \( (4) \) by 8 and equation \( (5) \) by 5 to eliminate one variable.

\[
8(5x + 6z) = 8(7) \quad \Rightarrow \quad 40x + 48z = 56
\]
\[
5(8x - 5z) = 5(-18) \quad \Rightarrow \quad 40x - 25z = -90
\]

### Step 4: Subtract the equations.

Subtract the second equation from the first:
\[
(40x + 48z) - (40x - 25z) = 56 - (-90)
\]
\[
40x + 48z - 40x + 25z = 56 + 90
\]
\[
73z = 146
\]
\[
z = 2
\]

### Step 5: Substitute \( z = 2 \) back into one of the earlier equations.

Substitute \( z = 2 \) into equation \( (4) \):
\[
5x + 6(2) = 7
\]
\[
5x + 12 = 7
\]
\[
5x = -5
\]
\[
x = -1
\]

### Step 6: Solve for \( y \).

Substitute \( x = -1 \) and \( z = 2 \) into \( y = 3x + z + 9 \):
\[
y = 3(-1) + 2 + 9
\]
\[
y = -3 + 2 + 9
\]
\[
y = 8
\]

### Final Solution:
The solution to the system of equations is:
\[
x = -1, \quad y = 8, \quad z = 2
\]

Do you want further details or have any questions? 

Here are 5 related questions for practice:
1. Solve the system of equations: \(2x + y - z = 4\), \(x - y + 3z = -1\), \(4x + 2y - z = 5\).
2. What is the determinant of the coefficient matrix of this system?
3. How would the solution change if the constant terms on the right-hand side were different?
4. Can you solve this system using matrix inversion?
5. What is the geometric interpretation of solving a system of three equations in three variables?

**Tip:** When solving systems of linear equations, eliminating variables through substitution or elimination is often the most direct method.

solve this system of equations: 8x - y + 7z = -2, 3x - y + z = -9, 5x + y - 6z = -9

Learn how to solve a system of linear equations step by step. This detailed solution demonstrates the substitution and elimination methods for solving 8x - y + 7z = -2, 3x - y + z = -9, and 5x + y - 6z = -9. Ideal for students in Grades 10-12 studying algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation