Problem 41 - Algebra 2

Question

Solve each matrix equation. \(\left[\begin{array}{l}{3 y} \\ {2 x}\end{array}\right]=\left[\begin{array}{l}{x+8} \\ {y-x}\end{array}\right]\)

Step-by-Step Solution

Answer

The solution is \(x = 1\) and \(y = 3\).

Step 1: Set Up Equations from the Matrices

Compare the corresponding elements of the matrices to form a system of equations. From the matrix equation, we have two equations:1. \(3y = x + 8\)2. \(2x = y - x\)

Step 2: Solve the First Equation for x

Start by solving the first equation \(3y = x + 8\) for \(x\). Rearrange it to express \(x\) in terms of \(y\):\[ x = 3y - 8 \]

Step 3: Substitute into the Second Equation

Now substitute \(x = 3y - 8\) into the second equation \(2x = y - x\):\[ 2(3y - 8) = y - (3y - 8) \]Simplify the equation.

Step 4: Simplify and Solve for y

Simplifying the substitution:\[ 6y - 16 = y - 3y + 8 \]Combine like terms:\[ 6y - 16 = -2y + 8 \]Add \(2y\) to both sides:\[ 8y - 16 = 8 \]Add \(16\) to both sides:\[ 8y = 24 \]Divide by 8:\[ y = 3 \]

Step 5: Substitute y into x Equation

Use the value \( y = 3 \) in the expression for \( x \):\[ x = 3(3) - 8 \]\[ x = 9 - 8 \]\[ x = 1 \]

Step 6: Conclusion

The solution to the matrix equation is \( x = 1 \) and \( y = 3 \).

Key Concepts

Understanding Systems of Equations

When we talk about a system of equations, we are dealing with a collection of two or more equations that share a common set of variables. Solving a system means finding the values for the variables that satisfy all the equations simultaneously. In this exercise, our system was derived from comparing corresponding elements in matrices. Each of these elements gave us one equation, resulting in our system of equations:

• Equation 1: \(3y = x + 8\)
• Equation 2: \(2x = y - x\)

Each equation describes a relationship between the variables \(x\) and \(y\). Our task is to find the values for \(x\) and \(y\) that satisfy both of these equations. This is a classic example of a system of linear equations.

Exploring the Substitution Method

The substitution method is a technique used to solve systems of equations. It involves solving one of the equations for one variable and then substituting that expression into the other equation. This reduces the system to a single equation with one variable.

In this problem, we start by rearranging the first equation to solve for \(x\):

• From \(3y = x + 8\), we isolate \(x\) to get \(x = 3y - 8\).

By substituting \(x = 3y - 8\) into the second equation \(2x = y - x\), we transformed it into an equation solely in terms of \(y\). This streamlines solving as we only have one variable to consider at a time. The substitution method is particularly useful in managing complex systems where direct computation of variables is cumbersome.

Simplifying Expressions in Equations

Simplifying expressions is a crucial step in solving equations as it makes calculations more manageable. After substitution in our exercise, the role of simplifying becomes apparent.

We inserted \(x = 3y - 8\) into the second equation, resulting in:

• \(2(3y - 8) = y - (3y - 8)\)

Simplification required us to distribute and combine like terms effectively:

• First, we expanded: \(6y - 16\) and \(-3y + 8\).
• Then, combined: \(6y - 16 = -2y + 8\).

Each step of simplification streamlines the problem down to its core elements, making it easier to identify the next necessary action in solving for the variable in question.

Solving Equations for Variable Values

After simplifying the equation, we focus on solving it to find the values of the unknowns. Using our simplified equation from the earlier step:

• \(8y - 16 = 8\)

We add \(16\) to both sides to isolate terms involving \(y\):

• This results in \(8y = 24\).

Finally, divide both sides by \(8\) to solve for \(y\):

• \(y = 3\).

With \(y\) known, substitute back into the equation for \(x\), derived from substitution:

• \(x = 3(3) - 8\), which simplifies to \(x = 1\).

Thus, the solutions found are \(x = 1\) and \(y = 3\), completing the solution of the matrix equation fully. This process highlights the systematic approach critical to solving any equation successfully.