https://file.aiquickdraw.com/mathbot/file/1731780655171qomu560p.jpg

The problem presents a system of linear equations:

\[
\frac{1}{2}x - y = -2
\]
\[
x + \frac{1}{3}y = 3
\]

We will solve it using the method of elimination.

### Step 1: Eliminate one variable
First, we rewrite the equations in a standard form to simplify elimination. Multiply through by the denominators to eliminate fractions.

1. Multiply the first equation by 2 to clear the fraction:
   \[
   x - 2y = -4
   \]

2. Multiply the second equation by 3 to clear the fraction:
   \[
   3x + y = 9
   \]

The system now becomes:
\[
x - 2y = -4
\]
\[
3x + y = 9
\]

### Step 2: Eliminate \(y\)
To eliminate \(y\), multiply the first equation by 1 (no change) and the second equation by 2:
\[
x - 2y = -4
\]
\[
6x + 2y = 18
\]

Add the equations together:
\[
(x - 2y) + (6x + 2y) = -4 + 18
\]
\[
7x = 14
\]
\[
x = 2
\]

### Step 3: Solve for \(y\)
Substitute \(x = 2\) into one of the original equations, e.g., \(x - 2y = -4\):
\[
2 - 2y = -4
\]
\[
-2y = -6
\]
\[
y = 3
\]

### Final Answer:
The solution to the system is:
\[
(x, y) = (2, 3)
\]

Let me know if you would like further clarification or details! Here are some related questions for further practice:

1. How would the solution change if one equation were altered slightly?
2. What is the geometric interpretation of solving a system of linear equations?
3. How does the elimination method compare to the substitution method?
4. Can you verify the solution graphically?
5. What happens if the system is inconsistent or dependent?

**Tip**: Always check your solution by substituting the values of \(x\) and \(y\) back into the original equations!

Solve the system by the method of elimination. (If there is no solution, enter NO SOLUTION. If the system is dependent, enter a for x and enter y in terms of a.)

Learn how to solve a system of linear equations using the elimination method. This step-by-step solution simplifies equations to find the values of x and y, ensuring clarity in every step. Suitable for Grades 8-10 students exploring 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