https://file.aiquickdraw.com/mathbot/file/1726068035985574tztcf.jpg

It seems you've uploaded a set of exercises from Section 3.2 involving solving systems of equations using Gaussian elimination. I'll go ahead and solve one of the equations for you and explain the steps in detail. Let me start with **Exercise 1**:

### Exercise 1:
The system of equations is:
\[
3x - 2y = 7
\]
\[
2x + 4y = 10
\]

We will solve this using the Gaussian elimination method:

### Step 1: Write the system in augmented matrix form
\[
\begin{pmatrix}
3 & -2 & | & 7 \\
2 & 4 & | & 10
\end{pmatrix}
\]

### Step 2: Perform row operations to get a row-echelon form
Our goal is to make the left side of the matrix an upper triangular matrix. Start by eliminating the \(x\)-term in the second row.

- Multiply row 1 by \(2\) and row 2 by \(3\) to align the coefficients of \(x\):
\[
\begin{pmatrix}
6 & -4 & | & 14 \\
6 & 12 & | & 30
\end{pmatrix}
\]
- Subtract row 1 from row 2:
\[
\text{Row 2} = \text{Row 2} - \text{Row 1}
\]
\[
\begin{pmatrix}
6 & -4 & | & 14 \\
0 & 16 & | & 16
\end{pmatrix}
\]

### Step 3: Solve for \(y\)
From row 2, we have:
\[
16y = 16 \implies y = 1
\]

### Step 4: Back-substitute to find \(x\)
Substitute \(y = 1\) into row 1:
\[
6x - 4(1) = 14 \implies 6x = 18 \implies x = 3
\]

### Solution:
The solution to the system is \(x = 3\), \(y = 1\).

---

Would you like me to solve another exercise or explain Gaussian elimination further? Here are five related questions to consider:

1. How does Gaussian elimination handle larger systems of equations?
2. What is the difference between Gaussian and Gauss-Jordan elimination?
3. How can we check if a system of equations has no solution or infinitely many solutions using matrices?
4. Can Gaussian elimination be applied to non-linear systems of equations?
5. How do row operations preserve the solution of a system?

**Tip:** Always check for consistency in the system after applying Gaussian elimination, as it helps detect any contradictions or special cases (no solutions or infinitely many).

Solve the system of equations using Gaussian elimination:
3x - 2y = 7
2x + 4y = 10

Learn how to solve the system of linear equations 3x - 2y = 7 and 2x + 4y = 10 using Gaussian elimination. This problem covers key concepts of algebra, including augmented matrix form and row reduction. Ideal for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation