Problem 55 - Algebra and Trigonometry Real Mathematics, Real People

Question

Matrix Multiplication Use matrix multiplication to determine whether each matrix is a solution of the system of equations. Use a graphing utility to verify your results. \(\left\\{\begin{aligned} x+2 y &=4 \\ 3 x+2 y &=0 \end{aligned}\right.\) (a) \(\left[\begin{array}{l}2 \\ 1\end{array}\right]\) (b) \(\left[\begin{array}{r}-2 \\ 3\end{array}\right]\) (c) \(\left[\begin{array}{r}-4 \\ 4\end{array}\right]\) (d) \(\left[\begin{array}{r}2 \\ -3\end{array}\right]\)

Step-by-Step Solution

Answer

The only solution to the system of equations is matrix (b) \(\left[\begin{array}{r}-2 \ 3\end{array}\right]\). The other matrices (a, c, and d) are not solutions to the system.

Step 1: Matrix (a)

Substitute \(x = 2\) and \(y = 1\) into the system of equations. The system becomes: \(2 + 2*1 = 4\) and \(3*2 + 2*1 = 0\). The first equation holds as \(4 = 4\), however the second equation does not as \(8 != 0\). Therefore, matrix (a) is not a solution.

Step 2: Matrix (b)

Substitute \(x = -2\) and \(y = 3\) into the system of equations. The system becomes: \(-2 + 2*3 = 4\) and \(3*(-2) + 2*3 = 0\). Both equations hold as they satisfy their respective sides, thus Matrix (b) is a solution.

Step 3: Matrix (c)

Substitute \(x = -4\) and \(y = 4\) into the system of equations. The system becomes: \(-4 + 2*4 = 4\) and \(3*(-4) + 2*4 = 0\). The first equation holds as \(4 = 4\), however, the second equation does not as \(-4 != 0\). Therefore, matrix (c) is not a solution.

Step 4: Matrix (d)

Substitute \(x = 2\) and \(y = -3\) into the system of equations. The system becomes: \(2 + 2*(-3) = 4\) and \(3*2 + 2*(-3) = 0\). Neither equations hold as they do not satisfy their respective sides. Therefore, matrix (d) is not a solution.

Key Concepts

Understanding System of Equations

A **system of equations** consists of two or more equations that share variables, and the goal is to find the values that satisfy all given equations simultaneously. In the given exercise, the system is made up of two linear equations: \[ \begin{aligned} x + 2y &= 4, \ 3x + 2y &= 0 \end{aligned} \] These equations share variables \(x\) and \(y\). Solving a system of equations involves finding pairs \((x, y)\) that make both equations true. Key methods to solve systems include:

• Graphical approach, where each equation is drawn on a graph and intersections represent solutions.
• Substitution method, focusing on solving one equation for one variable and replacing it in the other.
• Matrix multiplication, as used in this example, to determine if given matrices are solutions.

This problem asks if certain matrices correspond to solutions of the system.

Solution Verification in Systems

**Solution verification** is critical after finding potential solutions to confirm that they truly satisfy the entire system of equations. This process ensures accuracy and serves to eliminate mistakes. Here, the potential solutions are represented by matrices where the elements suggest specific \(x\) and \(y\) values. Two key steps to verify solutions include:

• Substitute the potential \(x\) and \(y\) values into each equation of the system.
• Check if both sides of the equations are equal, ensuring each equation holds true.

For instance, matrix (b) \( \left[\begin{array}{r}-2 \ 3\end{array}\right] \) was confirmed as a solution as both equations were satisfied when these values were substituted. Verification eliminates matrices that do not satisfy both equations fully.

Using a Graphing Utility

A **graphing utility** is a digital tool designed to plot equations and visually analyze solutions of systems. This helps in double-checking solutions derived through algebraic methods. Steps for using a graphing utility:

• Input each equation from the system and display their graphs.
• Observe the points where the lines intersect, representing the solution(s).
• Cross-verify these points with solutions found through other methods.

In this exercise, using a graphing utility can provide a visual assurance of the correctness of the solutions. Points of intersection on the graph should match the verified solutions from the algebraic process. This is an excellent way to bridge visual learning with analytical reasoning.

Exploring the Substitution Method

The **substitution method** is an effective tool for solving systems of equations, particularly when working with linear systems.Here’s how it typically works:

• Select one of the equations and solve for one of the variables in terms of the other. For example, from \(x + 2y = 4\), solve for \(x\) to get \(x = 4 - 2y\).
• Substitute this value into the other equation. Replace \(x\) in the second equation \(3x + 2y = 0\) with \(4 - 2y\): \ 3(4 - 2y) + 2y = 0
• Solve for the remaining variable \(y\). Plug this back into the first equation to solve for \(x\).

This method simplifies the process by reducing the number of variables to solve for, leading to quicker solutions. For systems represented by matrices, substitution helps pinpoint which matrices work as solutions.