Problem 17 - Algebra 2

Question

Find the value of each variable. $$ \left[\begin{array}{cc}{2} & {4} \\ {8} & {12}\end{array}\right]=\left[\begin{array}{cc}{4 x-6} & {-10 t+5 x} \\ {4 x} & {15 t+1.5 x}\end{array}\right] $$

Step-by-Step Solution

Answer

The value of the variables are \(x = 2\) and \(t = 0.6\).

Step 1: Set Up Matrix Equations

Each element in the corresponding position of the two matrices must be equal. Thus, we write down the following equations based on the matrix equality: 1) For the element in the first row and first column: \(2 = 4x - 6\).2) For the element in the first row and second column: \(4 = -10t + 5x\).3) For the element in the second row and first column: \(8 = 4x\).4) For the element in the second row and second column: \(12 = 15t + 1.5x\).

Step 2: Solve for x Using the Simplest Equation

From the third equation \(8 = 4x\), we can find the value of \(x\). To solve for \(x\), divide both sides by 4: \(x = \frac{8}{4} = 2\).

Step 3: Substitute x Into the Other Equations

Now substitute \(x = 2\) into the other equations:1) First equation becomes \(2 = 4(2) - 6\), which should be true if our solution for \(x\) is correct.2) Second equation becomes \(4 = -10t + 5(2)\) to solve for \(t\).3) Fourth equation becomes \(12 = 15t + 1.5(2)\), which also should be true if our solution for \(x\) is correct, but it's not necessary to use this for solving \(t\).

Step 4: Solve for t

We will now solve for \(t\) using the second equation with the substituted value of \(x\). The equation \(4 = -10t + 5(2)\) simplifies to \(4 = -10t + 10\). Subtract 10 from both sides to obtain \(-6 = -10t\) and then divide by -10 to solve for \(t\): \(t = \frac{-6}{-10} = 0.6\).

Key Concepts

Matrix Equality

Understanding matrix equality is critical when dealing with matrix equations. It simply means that two matrices are equal if and only if their corresponding elements are equal. In the context of the given exercise, we have two matrices:
\[\left[\begin{array}{cc}{2} & {4} \ {8} &{12}\end{array}\right]=\left[\begin{array}{cc}{4 x-6} & {-10 t+5 x} \ {4 x}& {15 t+1.5 x}\end{array}\right]\]To assert their equality, we set corresponding elements equal to each other, creating a set of linear equations to solve. This step is crucial as it transforms the problem from a matrix perspective to one involving individual algebraic equations, making the matrix equation solvable.

Systems of Linear Equations

When we talk about systems of linear equations, we refer to a collection of two or more linear equations involving the same set of variables. Our task is to find the values of these variables that satisfy all equations in the system. In this exercise, the matrix equality has given us a system of linear equations:

• 2 = 4x - 6
• 4 = -10t + 5x
• 8 = 4x
• 12 = 15t + 1.5x

We solve this system by finding the values for the variables x and t. One strategy is to isolate the simplest equation (#3 in this case) and find a value for one variable (x) before moving on to solve for the other (t). Each step in the solution process is critical for ensuring that all equations are correctly satisfied by the identified variable values.

Algebraic Substitution

The technique of algebraic substitution is a powerful tool in solving systems of equations. It involves solving one of the equations for one variable and then substituting that value into the other equations. In our exercise, solving the third equation (8 = 4x) for x gives us x = 2. We use this x value and substitute it back into the other equations to find t. By using substitution, we reduce the complexity of the system and make the other equations easier to solve. This method ensures consistency across the system and leads us to the correct solution.