Find the value of each variable.
$\left[\begin{array}{cc}
-5-t & 0 \\
8 & -22
\end{array}\right]=\left[\begin{array}{cc}
-5 & 0 \\
8 & -4 y-2
\end{array}\right]$
Answer (1)
Set up the equation − 5 − t = − 5 and solve for t .
Set up the equation − 22 = − 4 y − 2 and solve for y .
Find t = 0 .
Find y = 5 .
The solution is t = 0 , y = 5 .
Explanation
Setting up the problem We are given two matrices that are equal. This means that their corresponding entries must be equal. We can use this fact to set up equations and solve for the unknown variables t and y .
Solving for t The top-left entries give us the equation: − 5 − t = − 5
Value of t Adding 5 to both sides, we get: − t = 0 t = 0
Solving for y The bottom-right entries give us the equation: − 22 = − 4 y − 2
Isolating y Adding 2 to both sides, we get: − 20 = − 4 y
Value of y Dividing both sides by -4, we get: y = − 4 − 20 = 5
Final Answer Therefore, the values of the variables are t = 0 and y = 5 .
Examples
Imagine you're designing a game where two characters' stats need to be perfectly balanced. Each character's attributes (like strength, speed, and intelligence) can be represented in a matrix. If the characters are supposed to be equal in power, their matrices must be identical. By setting corresponding elements equal to each other, you ensure a fair and balanced gameplay experience. This is similar to how we solved for the variables t and y in the matrix equation.
