Problem 4 - College Algebra and Calculus: An Applied Approach

Question

Find \(x\) and \(y\). $$ \left[\begin{array}{ccc} x+2 & 8 & -3 \\ 1 & 2 y & 2 x \\ 7 & -2 & y+2 \end{array}\right]=\left[\begin{array}{ccc} 2 x+6 & 8 & -3 \\ 1 & 18 & -8 \\ 7 & -2 & 11 \end{array}\right] $$

Step-by-Step Solution

Answer

Therefore, the solution for the system of equations obtained from the given matrices are \(x = -4\) and \(y = 9\).

Step 1 - Equate Corresponding Elements

Equate the corresponding elements from both matrices. This will produce the following three equations: 1. \(x+2 = 2x + 6\)2. \(2y = 18\)3. \(y+2 = 11\)

Step 2 - Solve for \(y\)

Isolate \(y\) in the second equation by dividing both sides by 2. This will give: \(y = 18/2 = 9\). Do the same in the third equation and the result will also be \(y = 9\), hence validating the solution.

Step 3 - Substitute \(y\) into First Equation

Now that we know \(y = 9\), substitute this into the first equation: \(x+2 = 2x + 6\). Solve for \(x\) by first subtracting \(2x\) from both sides of the equation to get \(-x = 4\), and then multiplying by -1 to get \(x = -4\).

Key Concepts

Solving Systems of Equations

Solving systems of equations involves finding values for variables that make several equations true at the same time. In the exercise, each element of the matrices forms an equation that needs to be solved. This means the elements must equate to each other when they are in the same position in each matrix.
Let's outline some noteworthy aspects:

• Start by identifying which variables you need to solve for, here it's \(x\) and \(y\).
• Set up equations by equating corresponding elements from each matrix. For example, if the top-left corner of both matrices has different expressions, they must equal each other.
• You're essentially dealing with a system of linear equations. Solving these appropriately will yield the required solution.

When correctly equated, your set-up leads to a straightforward path of solving for the unknowns.

Matrix Equality

Matrix equality is a principle which states that two matrices are equal if they have the same dimensions and their corresponding elements are the same. This principle is crucial when solving matrix equations.

• The given matrices in the exercise need to be the same, meaning every element needs to match its corresponding counterpart in the other matrix.
• By identifying the elements that should be equal, you can form linear equations like the examples given in the solution \(x + 2 = 2x + 6\) and \(2y = 18\).
• Checking each element ensures that the solutions satisfy the equation throughout the entire matrix, not just in a particular position.

Matrix equality is your tool to break down the problem into manageable, solvable equations as compactly as possible.

Substitution Method

The substitution method is a classic technique for solving systems of equations. It's typically applied after isolating one variable in one equation, allowing you to substitute its value into another equation.
For this exercise:

• You start by solving one of the simpler equations from the matrix, such as \(2y = 18\).
• Once isolated to \(y = 9\), this value is substituted back into another more complex equation like \(x + 2 = 2x + 6\) to solve for another variable, \(x\).
• Substitution saves time and reduces the complexity, as it methodically breaks down and simplifies the equations one at a time.

This process illustrates how substitution can efficiently handle the task of solving variables, especially in systems extracted from equal matrices.