Problem 3 - College Algebra and Calculus: An Applied Approach

Question

Find \(x\) and \(y\). $$ \left[\begin{array}{rr} -4 & 3 \\ 6 & -1 \\ 8 & 2 \\ 5 & 9 \end{array}\right]=\left[\begin{array}{cc} x-2 & 3 \\ 6 & -1 \\ 8 & -x \\ 5 & 2 y-1 \end{array}\right] $$

Step-by-Step Solution

Answer

The solution for the equations is \(x = -2\) and \(y = 5\).

Step 1: Creating the Equations

Set the corresponding elements equal to each other to form the equations. Here we get two equations:1. The top left elements give the equation \(-4 = x-2\)2. The bottom right elements give the equation \(9 = 2y-1\).

Step 2: Solving for x

Solving the first equation \(-4 = x-2\), we get:Add 2 to both sides to isolate \(x\), we get \(x = -4 + 2 \), hence \(x = -2\).

Step 3: Solving for y

Solving the second equation \(9 = 2y-1\), we get:Add 1 to both sides to get \(2y = 9 + 1\) hence \(2y = 10\).Divide both sides by 2 to isolate \(y\), we get \(y = 10 / 2\), hence \(y = 5\).

Key Concepts

Matrix Equations

Matrix equations are a way to solve multiple equations at once using matrices. In this exercise, we're given two matrices that are equal. Each matrix is like a collection of numbers organized in rows and columns.
For our given problem, each entry in the matrix on the left must equal the same position entry on the right. This idea of matching corresponding elements is crucial when dealing with matrix equations. It allows us to take a complex system and break it down into simpler, manageable equations.

• Each pair of corresponding elements forms a separate equation.
• This approach turns a matrix problem into a system of linear equations.

Understanding matrix equations helps in simplifying and solving systems of equations, making complex problems much easier to handle.

Variable Isolation

Variable isolation is the process of rearranging an equation to find the value of a variable. It's one of the most important steps in solving equations.
In our exercise, after matching the elements of the matrices, we got two equations:

• (-4 = x-2) for the top left element.
• (9 = 2y-1) for the bottom right element.

To find the value of each variable (x and y), you need to "isolate" them.
Here’s how to do it:

• Move all other terms to the opposite side of the equation.
• For \(x\), add 2 to each side, which gives \(x = -2\).
• For \(y\), first add 1 to both sides, then divide by 2, arriving at \(y = 5\).

Making equations that isolate variables allows you to solve for unknowns step-by-step, ensuring you arrive at the correct solutions.

Matrix Equality

Matrix equality means that two matrices are equal if and only if all corresponding elements are equal. This concept makes it possible to set up equations from a given matrix problem.
In the given exercise, we used matrix equality by comparing entries from both matrices.
This means:

• Each entry in the first matrix matched with the corresponding entry in the second matrix.
• Creating equations such as (-4 = x-2) and (9 = 2y-1).

Understanding matrix equality helps simplify complex matrix problems. It breaks them into simple, related equations. This allows you to focus on matching individual elements to find unknown variables.