We can solve this system of linear equations using either substitution or elimination. I'll use the elimination method for this example:

The given system is:

\[
\begin{align*}
(1) \quad 2x - 5y &= 3 \\
(2) \quad 3x + 2y &= 14
\end{align*}
\]

### Step 1: Eliminate one variable
To eliminate one variable, let's make the coefficients of \( y \) in both equations the same. We can do this by multiplying equation (1) by 2 and equation (2) by 5:

\[
\begin{align*}
2(2x - 5y) &= 2(3) \quad \Rightarrow \quad 4x - 10y = 6 \quad \text{(3)} \\
5(3x + 2y) &= 5(14) \quad \Rightarrow \quad 15x + 10y = 70 \quad \text{(4)}
\end{align*}
\]

### Step 2: Add the two equations
Now, add equation (3) and equation (4) to eliminate \( y \):

\[
(4x - 10y) + (15x + 10y) = 6 + 70
\]

Simplifying:

\[
19x = 76
\]

### Step 3: Solve for \( x \)
Divide both sides by 19:

\[
x = \frac{76}{19} = 4
\]

### Step 4: Substitute \( x = 4 \) into one of the original equations
Now substitute \( x = 4 \) into equation (1):

\[
2(4) - 5y = 3
\]

This simplifies to:

\[
8 - 5y = 3
\]

Solving for \( y \):

\[
-5y = 3 - 8 = -5
\]

\[
y = \frac{-5}{-5} = 1
\]

### Final Answer:
The solution to the system is \( x = 4 \) and \( y = 1 \).

Would you like more details or have any questions?

Here are 5 related questions:
1. How can the substitution method be used to solve the same system?
2. What happens if both equations have the same slope in a system of linear equations?
3. How can graphing be used to solve systems of linear equations?
4. What is the determinant of a 2x2 system and how does it relate to solving it?
5. How do you know if a system of equations has no solution or infinitely many solutions?

**Tip:** When solving a system of equations, always check the solution by substituting the values back into both original equations.

Explore how to solve the system of linear equations 2x - 5y = 3 and 3x + 2y = 14 using the elimination method. This step-by-step solution simplifies both equations, eliminates variables, and solves for x and y. Ideal for students in Grades 8-10 studying algebra and systems of 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