Problem 17 - Algebra 2

Question

Solve each equation. $$ \left[\begin{array}{c}{4 x} \\\ {5}\end{array}\right]=\left[\begin{array}{l}{15+x} \\ {2 y-1}\end{array}\right] $$

Step-by-Step Solution

Answer

The solutions are \( x = 5 \) and \( y = 3 \).

Step 1: Understand Matrix Equality

Matrix equality states that two matrices are equal if and only if their corresponding elements are equal. This means for the given matrices \( \begin{bmatrix} 4x \ 5 \end{bmatrix} \) and \( \begin{bmatrix} 15+x \ 2y-1 \end{bmatrix} \), both elements in the first row and both elements in the second row must be equal.

Step 2: Set Up Equations from Matrix Equality

Based on the principle of matrix equality, equate the corresponding elements of the matrices.1. The equation from the first row is: \( 4x = 15 + x \).2. The equation from the second row is: \( 5 = 2y - 1 \).

Step 3: Solve for x

Solve the equation \( 4x = 15 + x \) for \( x \).Subtract \( x \) from both sides to obtain \( 4x - x = 15 \), which simplifies to \( 3x = 15 \).Divide both sides by 3 to find \( x \):\[ x = \frac{15}{3} = 5 \].

Step 4: Solve for y

Solve the equation \( 5 = 2y - 1 \) for \( y \).Add 1 to both sides to get \( 6 = 2y \).Divide both sides by 2 to find \( y \):\[ y = \frac{6}{2} = 3 \].

Step 5: Verify Solutions

Substitute \( x = 5 \) and \( y = 3 \) back into the equations to verify correctness:1. For \( 4x = 15 + x \): Check that \( 4(5) = 15 + 5 \), which simplifies to \( 20 = 20 \).2. For \( 5 = 2y - 1 \): Check that \( 5 = 2(3) - 1 \), which simplifies to \( 5 = 5 \).Both equations are satisfied, confirming the solutions are correct.

Key Concepts

Matrix Equality

When discussing matrix equality, we're looking at the condition that makes two matrices equal. For two matrices to be equal, each corresponding element must be identical. This means comparing entry by entry in each matrix.
For example, if we have two matrices:

• Matrix A: \( \begin{bmatrix} 4x \ 5 \end{bmatrix} \)
• Matrix B: \( \begin{bmatrix} 15 + x \ 2y - 1 \end{bmatrix} \)

Each entry in Matrix A must match the corresponding entry in Matrix B. In this context, matrix equality results in separate equations: \( 4x = 15 + x \) and \( 5 = 2y - 1 \). These are fundamental for solving the problem ahead.

System of Equations

A system of equations comes into play when we have more than one equation to solve simultaneously. From our matrices, we've translated the equality into two separate equations:

• Equation 1: \( 4x = 15 + x \)
• Equation 2: \( 5 = 2y - 1 \)

These equations need to be solved together because they share variables from the same problem setup. Solutions must satisfy all equations in the system simultaneously. Here, our goal is to find the values of \( x \) and \( y \) that work for both equations derived from the matrices.

Solving for Variables

To solve for the variables, we need to focus on each equation individually, while still considering them part of the larger system. Let's start with solving for \( x \):

• Equation 1: \( 4x = 15 + x \).
• Isolate \( x \) by subtracting \( x \) from both sides, yielding \( 3x = 15 \).
• Divide both sides by 3, resulting in \( x = 5 \).

Next, solve for \( y \):

• Equation 2: \( 5 = 2y - 1 \).
• Add 1 to both sides to get \( 6 = 2y \).
• Divide both sides by 2 to find \( y = 3 \).

Successfully finding \( x = 5 \) and \( y = 3 \) completes this step.

Verification of Solutions

Verification is crucial in ensuring the validity of the solutions found for our variables. We re-substitute \( x = 5 \) and \( y = 3 \) back into the original equations derived from matrix equality:

• Equation 1: Check if \( 4(5) = 15 + 5 \), which simplifies to \( 20 = 20 \). This confirms that our solution for \( x \) is correct.
• Equation 2: Check if \( 5 = 2(3) - 1 \), which simplifies to \( 5 = 5 \). This confirms that our solution for \( y \) is accurate.

By verifying, we ensure that the solutions not only meet the derived equations but maintain the integrity of the original problem statement. This step instills confidence that the calculations and logic applied were correct.