Solve the following system of equations for all three variables.
5, x, plus, 2, y, minus, z, equals, 10
5x+2y−z=
10
2, x, plus, 7, y, plus, 4, z, equals, minus, 9
2x+7y+4z=
−9
5, x, plus, 3, y, plus, z, equals, minus, 5
5x+3y+z=
−5

We are given the following system of three equations with three variables \(x\), \(y\), and \(z\):

1. \( 5x + 2y - z = 10 \)
2. \( 2x + 7y + 4z = -9 \)
3. \( 5x + 3y + z = -5 \)

We can solve this system using either substitution or elimination. I will use the **elimination method** for clarity.

### Step 1: Eliminate \(z\)
We will first eliminate \(z\) by manipulating the equations.

- From equation (1): \( 5x + 2y - z = 10 \)
  \[
  z = 5x + 2y - 10
  \]
  
- From equation (3): \( 5x + 3y + z = -5 \)
  Substitute \(z\) from equation (1):
  \[
  5x + 3y + (5x + 2y - 10) = -5
  \]
  Simplify:
  \[
  10x + 5y - 10 = -5
  \]
  \[
  10x + 5y = 5 \quad \text{(divide by 5)}
  \]
  \[
  2x + y = 1 \quad \text{(this is equation (4))}
  \]

### Step 2: Work with the new system
We now have:
1. \( z = 5x + 2y - 10 \)
2. \( 2x + y = 1 \) (equation (4))
3. \( 2x + 7y + 4z = -9 \) (equation (2))

Substitute \(z = 5x + 2y - 10\) into equation (2):
\[
2x + 7y + 4(5x + 2y - 10) = -9
\]
Expand:
\[
2x + 7y + 20x + 8y - 40 = -9
\]
Simplify:
\[
22x + 15y = 31
\]
\[
22x + 15y = 31 \quad \text{(this is equation (5))}
\]

### Step 3: Solve the system of two equations
We now have the system of two equations:
1. \( 2x + y = 1 \) (equation (4))
2. \( 22x + 15y = 31 \) (equation (5))

We can solve this using substitution or elimination. Let's use substitution.

From equation (4): 
\[
y = 1 - 2x
\]
Substitute this into equation (5):
\[
22x + 15(1 - 2x) = 31
\]
Expand:
\[
22x + 15 - 30x = 31
\]
Simplify:
\[
-8x = 16
\]
\[
x = -2
\]

### Step 4: Find \(y\) and \(z\)
- Substitute \(x = -2\) into \(y = 1 - 2x\):
  \[
  y = 1 - 2(-2) = 1 + 4 = 5
  \]
  
- Now substitute \(x = -2\) and \(y = 5\) into \(z = 5x + 2y - 10\):
  \[
  z = 5(-2) + 2(5) - 10 = -10 + 10 - 10 = -10
  \]

### Final Solution:
\[
x = -2, \quad y = 5, \quad z = -10
\]

Would you like a detailed explanation of any step? Let me know if you have any questions.

Here are 5 related questions:
1. How can the substitution method be used in systems of equations?
2. What are the advantages of the elimination method in solving linear systems?
3. How can matrix methods like Gaussian elimination solve systems of equations?
4. Can nonlinear systems of equations also be solved using elimination?
5. How do inconsistent systems of equations arise?

**Tip**: Always check your solution by substituting the values of \(x\), \(y\), and \(z\) back into the original equations to verify correctness.

Solve the following system of equations for all three variables: 5x + 2y - z = 10, 2x + 7y + 4z = -9, 5x + 3y + z = -5.

Learn how to solve a system of equations with three variables using the elimination method. The system includes the equations 5x + 2y - z = 10, 2x + 7y + 4z = -9, and 5x + 3y + z = -5. Suitable for students in Grades 9-12, this solution involves key algebraic techniques for solving linear equations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation