https://file.aiquickdraw.com/mathbot/file/1723035340812l1ikvudl.jpg

Let's go through each equation in the image and solve them step-by-step.

1. \( x + 3y = 2 \)
2. \( 2x - y = 1 \)
3. \( 3x + 2y = 10 \)
4. \( 2y - x = 3 \)
5. \( 2x - y + z = 3 \)
6. \( x + 3y + z = 6 \)
7. \( y - 2z = 1 \)
8. \( 3x + 2y - z = 7 \)

### Solution:

#### 1. \( x + 3y = 2 \)

\[
x = 2 - 3y
\]

#### 2. \( 2x - y = 1 \)

\[
2(2 - 3y) - y = 1 \\
4 - 6y - y = 1 \\
-7y = -3 \\
y = \frac{3}{7}
\]

Using \( y \) in the first equation to find \( x \):

\[
x + 3\left(\frac{3}{7}\right) = 2 \\
x + \frac{9}{7} = 2 \\
x = 2 - \frac{9}{7} \\
x = \frac{14}{7} - \frac{9}{7} \\
x = \frac{5}{7}
\]

So, \( x = \frac{5}{7} \) and \( y = \frac{3}{7} \).

#### 3. \( 3x + 2y = 10 \)

Using substitution for \( x \):

\[
3\left(\frac{5}{7}\right) + 2y = 10 \\
\frac{15}{7} + 2y = 10 \\
2y = 10 - \frac{15}{7} \\
2y = \frac{70}{7} - \frac{15}{7} \\
2y = \frac{55}{7} \\
y = \frac{55}{14}
\]

#### 4. \( 2y - x = 3 \)

Substitute \( y \):

\[
2\left(\frac{55}{14}\right) - \frac{5}{7} = 3 \\
\frac{110}{14} - \frac{10}{14} = 3 \\
\frac{100}{14} = 3 \\
y = \frac{15}{2}
\]

#### 5. \( 2x - y + z = 3 \)

Using substitution for \( x \) and \( y \):

\[
2\left(\frac{5}{7}\right) - \frac{15}{2} + z = 3 \\
\frac{10}{7} - \frac{105}{14} + z = 3 \\
z = 3 + \frac{105}{14} - \frac{10}{7}
\]

#### 6. \( x + 3y + z = 6 \)

Using \( x \) and \( y \):

\[
\frac{5}{7} + 3\left(\frac{15}{2}\right) + z = 6 \\
z = 6 - \frac{45}{2} - \frac{5}{7}
\]

#### 7. \( y - 2z = 1 \)

Substituting \( y \):

\[
\frac{15}{2} - 2z = 1 \\
z = \frac{7}{2}
\]

#### 8. \( 3x + 2y - z = 7 \)

Substituting \( x \) and \( y \):

\[
3\left(\frac{5}{7}\right) + 2\left(\frac{15}{2}\right) - \frac{7}{2} = 7
\]

### Summary:
This initial step-by-step attempt shows a consistent method of solving for \( x \), \( y \), and \( z \), although intermediate steps need verification. 

Would you like details or have any questions about these steps? 

Here are 8 related questions:
1. How can we verify the solutions to these linear equations?
2. What methods are available for solving systems of equations?
3. Can these solutions be represented graphically?
4. How do we solve systems with three variables?
5. What are the properties of consistent, inconsistent, and dependent systems?
6. How can matrices be used to solve these systems?
7. What is the role of determinant in solving linear systems?
8. How can substitution and elimination methods be compared?

Tip: Always check your solutions by substituting back into the original equations to ensure accuracy.

Learn how to solve systems of linear equations with three variables step-by-step. This problem involves using substitution and elimination methods to find solutions for x, y, and z. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation