Problem 14 - A Graphical Approach to College Algebra

Question

Find the value of each variable. Do not use a calculator. $$\left[\begin{array}{ccc} a+2 & 1 & 5 m \\ 8 k & 0 & 3 \end{array}\right]+\left[\begin{array}{ccc} 3 a & 2 z & 5 m \\ 2 k & 5 & 6 \end{array}\right]=\left[\begin{array}{ccc} 10 & -14 & 80 \\ 10 & 5 & 9 \end{array}\right]$$

Step-by-Step Solution

Answer

\(a = 2,\ z = -7.5,\ m = 8,\ k = 1\)

Step 1: Set Up the Equation for Each Element

Start by equating each corresponding element from the matrices on both sides of the equation. For the first element, we have:\((a+2) + 3a = 10\)For the second element:\(1 + 2z = -14\)For the third element:\(5m + 5m = 80\)For the fourth element:\(8k + 2k = 10\)Notice the middle element in the second row does not contain variables, so it verifies itself:\(0 + 5 = 5\)And for the last element:\(3 + 6 = 9\)

Step 2: Solve for 'a'

For the equation \((a+2) + 3a = 10\):Combine like terms:\(4a + 2 = 10\)Subtract 2 from both sides:\(4a = 8\)Divide by 4:\(a = 2\)

Step 3: Solve for 'z'

For the equation \(1 + 2z = -14\):Subtract 1 from both sides:\(2z = -15\)Divide by 2:\(z = -7.5\)

Step 4: Solve for 'm'

For the equation \(5m + 5m = 80\):Combine like terms:\(10m = 80\)Divide by 10:\(m = 8\)

Step 5: Solve for 'k'

For the equation \(8k + 2k = 10\):Combine like terms:\(10k = 10\)Divide by 10:\(k = 1\)

Key Concepts

Variables in Matrices

When dealing with matrices, each element in the matrix can potentially hold a variable. Understanding this concept is key to solving any matrix equation. Instead of dealing with variables in one-dimensional equations, you are now managing variables across the two dimensions of a matrix. Let's break this down:

• A matrix is like a box of numbers, organized in rows and columns.
• Each number in the matrix is called an element. Elements can be constants or variables.
• Variables are usually represented by symbols like \(a\), \(z\), \(m\), and \(k\) in this exercise.

When working with matrices, you often need to solve for these variables. This may seem daunting, as you're trying to find unknown quantities from a whole set of numbers. Think of it as solving several smaller problems within a bigger picture. Each variable is like a little puzzle piece that fits into the overall matrix equation.

Solving Algebraic Equations

Now that we understand what variables in matrices are, let's look at how to solve the algebraic equations that arise from them. The process of solving for a variable in a matrix involves similar steps to solving any other algebraic equation. Let's go over these steps:

• **Setting Up Equations:** Start by matching each corresponding element in the given matrices. This step is essential as it gives you individual equations to solve, like \((a+2) + 3a = 10\).
• **Isolating the Variable:** Simplify each equation to isolate the variable on one side. This often involves combining like terms and performing basic arithmetic operations such as addition, subtraction, multiplication, or division. For example, combining terms in the \(a\) equation gives \(4a + 2 = 10\).
• **Solving the Equation:** Finally, solve for the variable by reversing operations. Subtracting and dividing are common in the final steps to isolate the variable. From \(4a = 8\), dividing by 4 gives \(a = 2\).

Each of these steps brings you closer to finding the value of each variable. Mastery of these techniques will make tackling any matrix problem more straightforward and manageable.

Step-by-Step Solutions

Applying a step-by-step approach systematically breaks down the process, making a matrix equation less intimidating. Here's how this strategy is useful for solving matrix equations:

• **Identify Each Component:** Begin by identifying and writing down each individual algebraic equation from the corresponding elements of the matrices involved. It's like identifying what you know about each piece of the puzzle.
• **Solve Part by Part:** Focus on solving one equation at a time. This stops you from being overwhelmed with the full matrix and breaking it down into manageable parts. For example, solve the equation for \(a\) first, then move to others like \(z\), \(m\), and \(k\).
• **Check Work & Solution:** Upon solving all equations, substitute the values back into the original settings. This ensures that the solutions make sense and everything checks out. A double-check can save you from small mistakes that may lead to incorrect solutions.

Following these steps ensures you not only have the solution but understand the reasoning behind it. This methodical process is key to mastering matrix equations.