Problem 22 - Precalculus with Limits

Question

Solving a System of Linear Equations In Exercises \(19 - 24\) , solve the system of linear equations and check any solutions algebraically. $$\left\\{ \begin{aligned} x + 2 y & = 1 \\ 5 x - 4 y & = - 23 \end{aligned} \right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is \( x = -3 \) and \( y = 2 \).

Step 1: Multiply to get ready to eliminate

To get started with solving the system, we can prepare to eliminate one variable by multiplying the entire first equation by 4, and the entirety of the second equation by 2. This should give us a new system: \[ \left\{ \begin{aligned} 4x + 8y & = 4 \ 10x - 8y & = -46 \end{aligned} \right. \]

Step 2: Add the equations to eliminate y

By adding the two equations from Step 1 together, the term with y will cancel out, leaving us with a more manageable equation to solve. This results in \( 14x = -42 \)

Step 3: Solve for x

Divide the equation from Step 2 by 14 to solve for x. \( x = -42 / 14 = -3 \)

Step 4: Substitute x into original equation to solve for y

Substitute x = -3 back into the first original equation, \( -3 + 2y = 1 \), and solve for y \( 2y = 4 \), so \( y = 2 \)

Step 5: Check the Solution

To ensure that x = -3 and y = 2 are valid solutions for the system, you can substitute them into each original equation and see if both sides equal. The correct substitutions should result in valid equalities.

Key Concepts

Elimination Method

The Elimination Method is a popular technique used to solve systems of linear equations by removing one of the variables, enabling easier calculation of the remaining ones. This is typically done by adding or subtracting the equations once they've been adjusted appropriately.

In our solved example, the first step was to manipulate the equations to make elimination possible. Multiplying the first equation by 4 and the second by 2 transformed them into:

• \(4x + 8y = 4\)
• \(10x - 8y = -46\)

Adding these equations in Step 2 eliminated \(y\), resulting in the simpler equation \(14x = -42\). This process shows the power of the Elimination Method in simplifying complex equations.

Substitution Method

The Substitution Method involves solving one equation for one variable and then substituting that expression into another equation. This method effectively reduces the problem to one equation with one variable.

Once we found \(x = -3\) from the elimination process, we used substitution to find \(y\). Substituting \(x = -3\) into the original first equation \(x + 2y = 1\) converts it to \(-3 + 2y = 1\). Solving for \(y\) gives us \(y = 2\). While not the central method used here, substitution is key in finalizing solutions and cross-verifying results in other systems.

Solution Verification

After finding solutions, verifying them is crucial to confirm their correctness. This process is about plugging the solution pairs \((x, y)\) back into the original equations to ensure equality holds.

In our example, we found \(x = -3\) and \(y = 2\). Substituting into the first equation \(x + 2y = 1\), we calculate \(-3 + 2(2) = 1\), which is true. Similarly, the second equation \(5x - 4y = -23\) becomes \(5(-3) - 4(2) = -23\), also a valid equality.

Verification helps ensure no errors were made during computation, and assures that the solutions are accurate and reliable.

Linear Algebra

Linear Algebra is a foundational field of mathematics that studies vectors, vector spaces, and systems of linear equations. It provides tools and techniques to solve such systems efficiently and practically.

A system of linear equations can be represented as a matrix equation in the form \(AX = B\), where \(A\) is the coefficient matrix, \(X\) is the column matrix of variables, and \(B\) is the constants matrix.

The methods demonstrated, such as Elimination and Substitution, are direct approaches derived from Linear Algebra concepts, aimed at manipulating equations to find solutions. Understanding Linear Algebra empowers students to handle more complex algebraic structures and sets a foundation for computational algorithms used in technology and sciences.