Problem 19 - College Algebra

Question

Solve the system by Gaussian elimination. \(\left[\begin{array}{rr|r}-1 & 2 & -3 \\ 4 & -5 & 6\end{array}\right]\)

Step-by-Step Solution

Answer

The solution is \(x = -1\), \(y = -2\).

Step 1: Write the System of Equations

The given matrix represents a system of linear equations. From the matrix, we can write the system as below:\[\begin{align*}-1x + 2y &= -3, \4x - 5y &= 6.\end{align*}\]

Step 2: Eliminate x from the Second Equation

To eliminate \(x\) from the second equation, we can make the coefficient of \(x\) in both equations the same. We multiply the entire first equation by 4 to match the coefficient of \(x\) in the second equation:\[\begin{align*}4(-1x + 2y) &= 4(-3), \4x - 5y &= 6.\end{align*}\]This becomes:\[\begin{align*}-4x + 8y &= -12, \4x - 5y &= 6.\end{align*}\]

Step 3: Add the Equations

Now, add the transformed first equation and the second equation to eliminate \(x\):\[\begin{align*}(-4x + 8y) + (4x - 5y) &= -12 + 6.\end{align*}\]This simplifies to:\[\begin{align*}0x + 3y &= -6.\end{align*}\]Which results in:\[\begin{align*}3y &= -6.\end{align*}\]

Step 4: Solve for y

We solve the equation \(3y = -6\) by dividing both sides by 3:\[\begin{align*}y &= \frac{-6}{3} = -2.\end{align*}\]

Step 5: Substitute y into the First Equation

Substitute \(y = -2\) back into the first equation to solve for \(x\):\[\begin{align*}-1x + 2(-2) &= -3, \-1x - 4 &= -3.\end{align*}\]Add 4 to both sides:\[\begin{align*}-1x &= 1.\end{align*}\]

Step 6: Solve for x

Solve \(-1x = 1\) by dividing both sides by -1:\[\begin{align*}x &= \frac{1}{-1} = -1.\end{align*}\]Thus, \(x = -1\).

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. In our example, the variables are \(x\) and \(y\). Each equation describes a straight line when plotted on a graph, and solving the system means finding the point(s) where the lines intersect. For the matrix given, the system of equations derived can be:

• \(-1x + 2y = -3\)
• \(4x - 5y = 6\)

In simpler terms, these equations tell us how \(x\) and \(y\) are related to each other by combining different line slopes and intercepts.
When solving, we try to find values of \(x\) and \(y\) that satisfy both equations simultaneously. Successfully finding these values effectively solves the system.
This is important not only in mathematics but also in many practical fields like physics and engineering, where multiple variables need to meet specific conditions.

Matrix Representation

Matrix representation is a compact way to express a system of linear equations. Instead of writing out every individual equation, a matrix provides a neat and organized array of numbers, representing the coefficients and constants of each equation. For instance, the matrix: \[\left(\begin{array}{rr|r}-1 & 2 & -3 \4 & -5 & 6\end{array}\right)\]Emphasizes the coefficients \(-1, 2, 4,\) and \(-5\) for the variables \(x\) and \(y\), alongside their respective constant values \(-3\) and \(6\). This setup is particularly useful in linear algebra because it allows for the application of systematic methods like Gaussian elimination to solve the system.
The process of converting equations into a matrix is straightforward:

• List coefficients of each variable in each row, corresponding to each equation.
• Place constants after a dashed line in augmented form.

This approach simplifies handling multiple equations at once, ensuring calculations are more organized.

Elimination Method

Gaussian elimination, or the elimination method, is a step-by-step process used to solve systems of linear equations. The goal is to systematically eliminate variables to eventually solve for each one. Here's how it works using our example system:**Step 1:** Multiply or scale equations to equalize coefficients. Here, the first equation \(-1x + 2y = -3\) is multiplied by 4, resulting in \(-4x + 8y = -12\), aligning it with the second equation \(4x - 5y = 6\).
**Step 2:** Add or subtract equations to eliminate variables. The process subtracts the equations, leading to cancellation of the \(x\) terms, simplifying it to \(3y = -6\).
**Step 3:** Solve the result. Once a single variable equation is left, it's straightforward to solve, as with \(3y = -6\), where dividing both sides by 3 gives \(y = -2\).
**Step 4:** Substitute back to find remaining variables. Substituting \(y = -2\) back into one of the original equations helps find \(x\). Substituting into \(-1x + 2(-2) = -3\) simplifies to \(x = -1\).
By visualizing each row operation and substitution, students can better understand how changes to one equation affect the overall system, leading to the ultimate solution for \(x\) and \(y\). This method is powerful, especially for larger systems, as it systematically leads to the solution with clarity.