Problem 18 - Algebra 2

Question

Solve each equation. $$ \left[\begin{array}{l}{x+3 y} \\ {3 x+y}\end{array}\right]=\left[\begin{array}{r}{-13} \\ {1}\end{array}\right] $$

Step-by-Step Solution

Answer

The solutions are \( x = 2 \) and \( y = -5 \).

Step 1: Set Up the Equations

From the given matrix equation, extract two separate linear equations. The matrix equation \( \left[ \begin{array}{l} x+3y \ 3x+y \end{array} \right] = \left[ \begin{array}{r} -13 \ 1 \end{array} \right] \) represents two equations: \( x + 3y = -13 \) and \( 3x + y = 1 \).

Step 2: Solve for y in Equation Two

Using the second equation \( 3x + y = 1 \), solve for \( y \). Rearrange it to get \( y = 1 - 3x \).

Step 3: Substitute y into Equation One

Take the expression for \( y \) from Step 2 and substitute it into the first equation \( x + 3y = -13 \). This gives: \( x + 3(1 - 3x) = -13 \).

Step 4: Simplify and Solve for x

Simplify the equation from Step 3: \( x + 3 - 9x = -13 \), which simplifies to \( -8x + 3 = -13 \). Further simplify to solve for \( x \): \( -8x = -16 \), so \( x = 2 \).

Step 5: Substitute x back to solve for y

Substitute \( x = 2 \) back into the expression for \( y \) from Step 2: \( y = 1 - 3(2) = 1 - 6 = -5 \).

Step 6: Verify the Solutions

Check the solutions \( x = 2 \) and \( y = -5 \) in the original equations. For \( x + 3y = -13 \), substituting gives \( 2 + 3(-5) = 2 - 15 = -13 \) (correct). For \( 3x + y = 1 \), substituting gives \( 3(2) + (-5) = 6 - 5 = 1 \) (correct).

Key Concepts

Matrix Representation

Matrix representation is a powerful way to express a system of linear equations. Instead of writing each equation separately, you can organize them into a matrix format. For our example, we started with the matrix equation: \[ \left[ \begin{array}{l} x+3y \ 3x+y \end{array} \right] = \left[ \begin{array}{r} -13 \ 1 \end{array} \right] \] The first column typically contains the variables, and the second column contains the constants. This matrix condenses both equations:

• \( x + 3y = -13 \)
• \( 3x + y = 1 \)

By organizing them in this way, you can easily visualize the entire system. It's also a first step towards other sophisticated techniques like using the determinant or inverse of matrices to find solutions.

Substitution Method

The substitution method is a systematic procedure for solving systems of equations. It involves expressing one of the variables in terms of the others, and then substituting this expression into another equation. In our exercise, we began by isolating \( y \) in one of the equations. By taking the second equation \( 3x + y = 1 \) and rearranging it, we found: \( y = 1 - 3x \). This expression allows us to substitute \( y \) in the first equation: \( x + 3(1 - 3x) = -13 \)
Rearranging and simplifying this, we focus on solving for \( x \) and find that \( x = 2 \). This value is then substituted back in the expression for \( y \), resulting in \( y = -5 \). The substitution method is especially useful in cases where one variable is easy to isolate, which simplifies finding the solution for the system.

Verification of Solutions

Verification is essential to ensure that the calculated answers are correct. After obtaining the solutions \( x = 2 \) and \( y = -5 \), we substitute these values back into the original equations to confirm their validity. Starting with \( x + 3y = -13 \), substituting gives: \( 2 + 3(-5) = 2 - 15 = -13 \). This confirms that our solutions satisfy the first equation.
Next, we check \( 3x + y = 1 \). Substituting gives: \( 3(2) + (-5) = 6 - 5 = 1 \), which confirms the correct solution for the second equation. By verifying solutions, you ensure that no errors occurred during the manipulation of the equations. It's like double-checking your work, an essential part of the problem-solving process, providing confidence in your mathematical reasoning.