Problem 35 - A Graphical Approach to College Algebra

Question

Use row operations on an augmented matrix to solve each system of equations. Round to nearest thousandth when appropriate. $$\begin{aligned} &2 x-3 y=10\\\ &2 x+2 y=5 \end{aligned}$$

Step-by-Step Solution

Answer

The solution is \(x = 3.5\), \(y = -1\).

Step 1: Write the Augmented Matrix

First, write the system of linear equations as an augmented matrix. For the given system \(2x - 3y = 10\) and \(2x + 2y = 5\), the augmented matrix is: \[\begin{bmatrix} 2 & -3 & | & 10 \2 & 2 & | & 5 \end{bmatrix}\]

Step 2: Perform Row Operations to Eliminate x

To eliminate the \(x\) variable from the second row, subtract the first row from the second row:\[R_2 = R_2 - R_1\]This gives the new augmented matrix:\[\begin{bmatrix} 2 & -3 & | & 10 \0 & 5 & | & -5 \end{bmatrix}\]

Step 3: Solve for y

Use the second row of the matrix \(0y + 5y = -5\) to solve for \(y\). By simplifying, we find:\[5y = -5\]Divide both sides by 5:\[y = -1\]

Step 4: Substitute y Back to Find x

Substitute \(y = -1\) into one of the original equations, for example, \(2x - 3y = 10\):\[2x - 3(-1) = 10\]Simplify and solve for \(x\):\[ 2x + 3 = 10\]\[2x = 7\]Divide both sides by 2:\[x = 3.5\]

Step 5: Verify the Solution

Substitute \(x = 3.5\) and \(y = -1\) back into the second original equation to verify the solution:\[2(3.5) + 2(-1) = 5\]\[7 - 2 = 5\]This is true, confirming that the solution is correct.

Key Concepts

Understanding Row Operations

Row operations are crucial when working with augmented matrices to solve systems of equations. They allow us to manipulate the rows of a matrix to achieve a desired form without actually changing the solution to the system. This is just like solving the equations by hand but in a structured and organized way.There are three primary row operations:

• Swapping two rows
• Multiplying a row by a non-zero constant
• Adding or subtracting the multiples of one row to another row

In the given example, we use the operation of subtracting the first row from the second one to eliminate the variable \(x\) from the second row.
Just as with solving equations by hand, these operations help simplify and progress towards the solution.

Solving Systems of Equations

A system of equations involves finding the values of variables that satisfy multiple equations at once. In our example, we have two equations with two unknowns: \(x\) and \(y\). Systems of equations can generally be solved through:

• Substitution
• Elimination
• Graphical methods
• Using matrices and row operations

Using an augmented matrix along with row operations helps in simplifying and solving the system efficiently. By transforming the matrix, we reduce the system to a simpler form from which the solution can be easily obtained.

The Basics of Linear Equations

Linear equations are equations where the variables are in the first degree, meaning they are not squared, cubed, etc. They appear in the form \(ax + by = c\). In our example, such linear equations are \(2x - 3y = 10\) and \(2x + 2y = 5\).Important points about linear equations:

• Each solution represents a point on a coordinate plane where the two lines intersect
• If two lines are parallel, they have no points in common, leading to no solution
• If two lines coincide, there are infinitely many solutions

Linear equations are the foundational elements within algebra upon which many complex topics are built.

Matrix Solutions and Their Importance

Matrices are an organized way of representing and solving equations, especially useful for systems with multiple equations. With an augmented matrix, we're incorporating both the coefficients of variables and the constants—everything needed to solve the system into one table-like array. Using augmented matrices to find solutions provides several benefits:

• Reduces computational error by structuring the process
• Makes it easier to handle large systems of equations
• Facilitates using computers and calculators for solving complex systems efficiently

Understanding matrix solutions involves familiarizing oneself with terms like pivot, row-echelon form, and reduced row-echelon form. The matrix solution method is a gateway to more advanced mathematical concepts such as linear transformations and eigenvalues.