Problem 48 - Algebra 2

Question

ACT/SAT Solve for \(x\) and \(y\) in the matrix equation \(\left[\begin{array}{l}{x} \\ {7}\end{array}\right]+\left[\begin{array}{c}{3 y} \\ {-x}\end{array}\right]=\left[\begin{array}{c}{16} \\\ {12}\end{array}\right]\) A \(x=-5, y=7\) B \(x=7, y=3\) C \(x=7, y=5\) D \(x=5, y=7\)

Step-by-Step Solution

Answer

The correct answer is A: \(x = -5, y = 7\).

Step 1: Set Up the Matrix Equations

We begin by identifying the components in the given matrix equation. We have two vectors being added:\[\begin{bmatrix} x \ 7 \end{bmatrix} + \begin{bmatrix} 3y \ -x \end{bmatrix} = \begin{bmatrix} 16 \ 12 \end{bmatrix}.\] This sets up two equations by equating corresponding components of the matrices.

Step 2: Equating the First Components

From the first row of the matrix equation, we equate the elements:\[x + 3y = 16.\] This is our first equation relating \(x\) and \(y\).

Step 3: Equating the Second Components

From the second row of the matrix equation, we get:\[7 - x = 12.\] Solving for \(x\), we find that \(x = 7 - 12\). Thus, \(x = -5\).

Step 4: Solve for \(y\) Using Equation 1

Substitute \(x = -5\) into the first equation:\[-5 + 3y = 16\] Solve for \(y\) by adding 5 to both sides, which gives us:\[3y = 21\] Finally, divide both sides by 3:\[y = 7.\]

Step 5: Verify the Solution

We can verify that \(x = -5\) and \(y = 7\) satisfy both original equations:1. For the first equation, substituting gives: \[-5 + 3(7) = 16 \] (True)2. For the second equation: \[7 - (-5) = 12 \] (True) Both equations hold, confirming our solution.

Key Concepts

Solving Systems of Equations

Solving systems of equations often feels like solving a puzzle. Here, our puzzle is embedded within a matrix equation which offers a powerful way to encapsulate multiple linear equations succinctly. The essence of solving such a system lies in finding values of variables that satisfy all given conditions. In our case, we have a system with two equations: one coming from equating the first row of matrices and the other from the second. Each element in the matrix corresponds directly to an equation, simplifying the complexity of handling multiple variables:

• First, we equate the elements of the first row to form the equation: \( x + 3y = 16 \).
• Then, we use the second row to form another equation: \( 7 - x = 12 \).

By solving these two equations systematically, we determine the values of \(x\) and \(y\) such that both conditions hold true. Such a process develops critical analytical and algebraic skills.

Matrix Addition

Matrix addition is a simple yet foundational concept in linear algebra and comes into play when solving matrix equations. When adding two matrices, we operate element-wise, meaning we add corresponding elements together. This makes matrix addition straightforward as long as the matrices are compatible, i.e., they have the same dimensions (same number of rows and columns). During our example, we added two vectors, effectively a 1x2 matrix, using straightforward arithmetic:

• Add the top elements: \( x + 3y \).
• Add the bottom elements: \( 7 - x \).

The result of this addition is a new vector: \( \begin{bmatrix} 16 \ 12 \end{bmatrix} \). Matrix addition emphasises understanding of pairwise operations, and provides a visual and structural method to tackle algebraic problems, which is particularly useful in higher mathematical applications.

ACT/SAT Math Problems

Solving matrix equations is a valuable skill for ACT and SAT math problems. These standardized tests often challenge students to think critically and apply basic mathematical concepts in novel contexts. Matrix equations, for example, are not just about standard addition or solving linear equations, but combine several concepts in one problem which appeals to problem-solving and logical reasoning skills. For matrix-related problems:

• First, identify what each component of the matrix represents.
• Translate these into separate equations, as we did here by matching corresponding elements to form two equations.

Practicing such exercises hones your ability to see mathematical problems from different perspectives. Completing these tasks underpinned by contexts similar to those in ACT/SAT problems prepares students well by reinforcing their understanding of algebra, geometry, and functions integrated in a single task.