Problem 12 - A Graphical Approach to College Algebra

Question

Find the value of each variable. Do not use a calculator. $$\left[\begin{array}{ccc} 3+x & 4 & t \\ 5 & 8-w & y+1 \\ -4 & 3 & 2 r \end{array}\right]=\left[\begin{array}{ccc} 9 & 4 & 6 \\ z+3 & w & 9 \\ p & q & r \end{array}\right]$$

Step-by-Step Solution

Answer

x = 6, w = 4, t = 6, z = 2, y = 8, p = -4, q = 3, r = 0.

Step 1: Understand Matrix Equality

Matrix equality means that corresponding elements of two matrices are equal. We are given two 3x3 matrices, and their corresponding elements must match. This gives us a system of equations for each pair of matching elements.

Step 2: Setting Up Equations for x

Look at the element in the first row, first column: \(3 + x\) must equal \(9\). Thus, we have the equation: \(3 + x = 9\).

Step 3: Solve for x

Subtract 3 from both sides of the equation \(3 + x = 9\) to isolate \(x\) on one side. This yields \(x = 6\).

Step 4: Setting Up Equations for w

Look at the element in the second row, second column: \(8 - w\) must equal \(w\). Thus, we have the equation: \(8 - w = w\).

Step 5: Solve for w

Add \(w\) to both sides of the equation \(8 - w = w\). This simplifies to \(8 = 2w\). Divide both sides by 2 to isolate \(w\), resulting in \(w = 4\).

Step 6: Setting Up Equations for t

Look at the element in the first row, third column: \(t\) must equal \(6\). Thus, \(t = 6\) is already determined.

Step 7: Setting Up Equations for z

Look at the element in the second row, first column: \(z + 3\) must equal \(5\). Thus, we have the equation: \(z + 3 = 5\).

Step 8: Solve for z

Subtract 3 from both sides of the equation \(z + 3 = 5\). This gives \(z = 2\).

Step 9: Setting Up Equations for y

Look at the element in the second row, third column: \(y + 1\) must equal \(9\). Thus, we have the equation: \(y + 1 = 9\).

Step 10: Solve for y

Subtract 1 from both sides of the equation \(y + 1 = 9\). This results in \(y = 8\).

Step 11: Setting Up Equations for p

Look at the element in the third row, first column: \(-4\) must equal \(p\). Thus, \(p = -4\) is already determined.

Step 12: Setting Up Equations for q

Look at the element in the third row, second column: \(3\) must equal \(q\). Thus, \(q = 3\) is already determined.

Step 13: Setting Up Equations for r

Look at the element in the third row, third column: \(2r\) must equal \(r\). Thus, we have the equation: \(2r = r\).

Step 14: Solve for r

Subtract \(r\) from both sides of the equation \(2r = r\). This results in \(r = 0\).

Key Concepts

Matrix equality

Matrix equality is a fundamental concept in linear algebra that means two matrices are considered equal if and only if their corresponding elements are identical. In our exercise, we have two 3x3 matrices, and to determine their equality, each pair of corresponding elements from both matrices must match.

• For example, the element in the first row and first column in the first matrix must be equal to the element in the same position of the second matrix.
• This comparison leads us to form a system of equations that can be solved to find the unknown variables.

Matrix equality is important in many applications, such as solving simultaneous equations and in computer graphics, where it assists in operations like transformations and rotations.

System of equations

When dealing with matrix equality, you often end up creating a system of equations. This occurs because each pair of corresponding elements gives rise to an equation. In our exercise, after inspecting the given matrices, we established several equations, such as:

• 3 + x = 9
• 8 - w = w
• z + 3 = 5
• y + 1 = 9

Each equation can be solved separately for its respective variable. The challenge is to correctly set all equations so that the final solution satisfies the conditions of matrix equality.

Variable solving

Variable solving involves isolating the variable in an equation to determine its value. This step is crucial in linear algebra, especially when finding unknowns in a matrix equation.
To solve for a variable, follow these steps: 1. Look at the equation. For example, consider 3 + x = 9. 2. Perform algebraic operations to isolate the variable. For 3 + x = 9, you would subtract 3 from both sides to find x = 6. 3. Repeat this process for each variable.
Additional Step: Always double-check your work by plugging the found value back into the original equation to ensure both sides are equal. This confirms that the variable has been correctly solved.

Linear algebra

Linear algebra is a branch of mathematics focusing on vectors, vector spaces, and linear transformations. It provides the framework for understanding concepts like matrix equality and systems of equations, which are core topics in the study of linear equations and matrix theory.
In our exercise:

• We applied linear algebra principles to set up equations for each matrix element.
• The process of solving these equations involves using basic algebraic techniques, such as addition, subtraction, and division, to find unknown variable values.

Linear algebra is powerful in real-world applications, including computer science, engineering, and physics. It helps solve problems involving multiple variables and complex calculations efficiently.