Solve the system below using augmented matrix methods. Graph each solution set. Discuss the difference between the graph of an equation in the system and the graph of the​ system's solution set.
5x 1negative 3x 2equals
20
3x 1negative 2x 2equals
10


We are given the system of linear equations:

\[
\begin{align*}
5x_1 - 3x_2 &= 20 \\
3x_1 - 2x_2 &= 10
\end{align*}
\]

### Step 1: Write the augmented matrix
The system can be rewritten as an augmented matrix, where the coefficients of \(x_1\), \(x_2\), and the constants are written in matrix form:

\[
\begin{pmatrix}
5 & -3 & | & 20 \\
3 & -2 & | & 10
\end{pmatrix}
\]

### Step 2: Perform row operations to obtain row-echelon form
We aim to use Gaussian elimination to simplify the augmented matrix into an easier form for back substitution.

1. **First, use the first row to eliminate the \(x_1\) term from the second row**.
   Multiply the first row by \(3\) and the second row by \(5\), then subtract the second row from the first:

\[
R_2 = 5R_2 - 3R_1
\]

\[
\begin{pmatrix}
5 & -3 & | & 20 \\
0 & -1 & | & -10
\end{pmatrix}
\]

2. **Next, make the second row's second element equal to 1**. Multiply the second row by \(-1\):

\[
R_2 = -1 \cdot R_2
\]

\[
\begin{pmatrix}
5 & -3 & | & 20 \\
0 & 1 & | & 10
\end{pmatrix}
\]

3. **Now, use the second row to eliminate the \(x_2\) term from the first row**:
   Multiply the second row by \(3\) and add it to the first row:

\[
R_1 = R_1 + 3R_2
\]

\[
\begin{pmatrix}
5 & 0 & | & 50 \\
0 & 1 & | & 10
\end{pmatrix}
\]

### Step 3: Solve the system
From the augmented matrix, we have:

\[
5x_1 = 50 \quad \Rightarrow \quad x_1 = \frac{50}{5} = 10
\]
\[
x_2 = 10
\]

Thus, the solution to the system is \(x_1 = 10\) and \(x_2 = 10\), or in coordinate form: \((10, 10)\).

### Step 4: Graph the solution set
Each equation in the system represents a line on a coordinate plane. The solution to the system is the point where the two lines intersect.

1. The first equation is \(5x_1 - 3x_2 = 20\), which is a line.
2. The second equation is \(3x_1 - 2x_2 = 10\), which is another line.

When graphed, these two lines will intersect at the point \((10, 10)\), which is the solution to the system.

### Graph of individual equations versus solution set
- **Graph of an individual equation**: Each equation represents a line on the coordinate plane. Every point on that line is a solution to that equation.
- **Graph of the system's solution set**: The solution to the system is the point where both lines intersect, which represents the pair of values that satisfies both equations simultaneously. In this case, the solution is the point \((10, 10)\).

### Do you need a detailed graph or have any more questions?

#### 5 Related Questions:
1. What are the steps to convert a system of equations to an augmented matrix?
2. How do row operations affect the augmented matrix?
3. What does it mean when two lines in a system of equations are parallel?
4. How can we interpret the solution of a system graphically when there is no solution?
5. How is Gaussian elimination different from Gauss-Jordan elimination?

#### Tip:
When performing Gaussian elimination, always aim to simplify each row step by step, ensuring that the matrix gets closer to upper triangular form.

Solve the system below using augmented matrix methods. Graph each solution set. Discuss the difference between the graph of an equation in the system and the graph of the​ system's solution set. 5x_1 - 3x_2 = 20, 3x_1 - 2x_2 = 10

Learn how to solve the system of linear equations 5x1 - 3x2 = 20 and 3x1 - 2x2 = 10 using augmented matrix methods. This detailed step-by-step guide explains Gaussian elimination, row operations, and graphing the solution set. Suitable for high school students in grades 10-12 or early college.

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