Problem 29 - College Algebra

Question

For the following exercises, solve the system by Gaussian elimination. $$ \begin{aligned} 2 x-y &=2 \\ 3 x+2 y &=17 \end{aligned} $$

Step-by-Step Solution

Answer

(3, 4)

Step 1: Write the Augmented Matrix

First, rewrite the system of equations as an augmented matrix. The given system is: \[\begin{align*}2x - y &= 2 \3x + 2y &= 17\end{align*}\] This converts to the matrix form: \[\begin{bmatrix}2 & -1 & \vline & 2 \3 & 2 & \vline & 17\end{bmatrix}\]

Step 2: Eliminate First Column of Second Row

To eliminate the element in the first column of the second row, multiply the first row by 3 and the second row by 2, then subtract the first row from the second row. Multiply: \(\begin{array}{c}2\text{Row }1: \ 6x - 3y = 6 \3\text{Row }2: \ 6x + 4y = 34\end{array}\)Subtract: \(6x + 4y - (6x - 3y) = 34 - 6\)Results in:\(7y = 28\)

Step 3: Solve for y

From the equation obtained in Step 2: \(7y = 28\). Solve for \(y\) by dividing both sides by 7:\[y = \frac{28}{7} = 4\]

Step 4: Back Substitute to Solve for x

Now that we have \(y = 4\), substitute this value back into the first equation of the original system, \(2x - y = 2\).Substitute: \(2x - 4 = 2\)Solve for \(x\): \(2x = 6\)\(x = 3\)

Step 5: Write Solution as Ordered Pair

The solutions obtained are \(x = 3\) and \(y = 4\). Therefore, the solution to the system of equations is the ordered pair \((3, 4)\).

Key Concepts

Augmented Matrix

In mathematics, when solving systems of equations, the augmented matrix is a convenient tool for applying the Gaussian Elimination method. It represents the entire system in a simple matrix form by including coefficients of the variables and constant terms.
Typically, each row in an augmented matrix corresponds to an equation in the system, while each column represents the coefficients of the different variables. For example, in the system:

• \( 2x - y = 2 \)
• \( 3x + 2y = 17 \)

The augmented matrix is written as:\[\begin{bmatrix}2 & -1 & \vline & 2 \3 & 2 & \vline & 17\end{bmatrix}\]This matrix includes a vertical bar separating the coefficients on the left from the constants on the right. This choice of format helps visualize the transition from algebraic equations to a matrix, setting the stage for further solving steps.

System of Equations

A system of equations is a collection of two or more equations with the same set of variables. The key challenge here is to find solutions that satisfy all equations simultaneously.
In our example, we have the following system:

• \( 2x - y = 2 \)
• \( 3x + 2y = 17 \)

To solve such systems, we employ methods like substitution, elimination, or Gaussian elimination. Importantly, solving a system corroborates the need for ordered solutions where each variable has a specific value that makes all equations true.
Conceptualize systems of equations as intersections of lines when plotted graphically. Each solution corresponds to a point where these lines meet, illustrating why multiple variables and equations lead to a single point of intersection.

Back Substitution

Once you partially solve the system of equations using Gaussian elimination, back substitution is the next step. It allows you to find the values of the remaining variables in reverse order.
Having transformed the augmented matrix into an upper triangular form, you start solving from the last equation upward. In our case, we found from the row operations that:

• \( 7y = 28 \)
• Solved to \( y = 4 \)

With this known value, return to one of the earlier equations to solve for \( x \). This is known as back substitution because it involves going back through the previously simplified matrix to substitute and solve for remaining variables.
This step ensures that all solutions comply with the original system, maintaining the integrity of the solution process.

Ordered Pair Solution

After following the process of Gaussian elimination and back substitution, it is customary to express the solution of a system of equations as an ordered pair. This uses the form \((x, y)\), where each element represents one of the variables.
In our exercise, we found:

• \( x = 3 \)
• \( y = 4 \)

Thus, the ordered pair that solves the system is \((3, 4)\). This format is simple, straightforward, and gives a clear numerical solution to the problem.
Ordered pairs are especially useful for plotting on a graph. By illustrating where the solutions lie in a coordinate plane, they provide visual confirmation that solutions satisfy the original set of equations, representing the prevalent intersection of all involved equations.