If $ \begin{bmatrix} 5x + 8 & 7 \ y + 3 & 10x + 12 \end{bmatrix} = \begin{bmatrix} 2 & 3y + 1 \ 5 & 0 \end{bmatrix} $ then the value of $ 5x + 3y $ is equal to:

Question

Prepp Team · Accepted Answer

Understanding Matrix Equality

When two matrices are stated to be equal, it means that each element in the first matrix is equal to the corresponding element in the second matrix. For two matrices to be equal, they must have the same dimensions (same number of rows and columns), and their elements in the same positions must be identical.

The given matrix equation is:

$$ \begin{bmatrix} 5x + 8 & 7 \\ y + 3 & 10x + 12 \end{bmatrix} = \begin{bmatrix} 2 & 3y + 1 \\ 5 & 0 \end{bmatrix} $$

Since these two matrices are equal, we can equate their corresponding elements. This gives us a system of linear equations involving the variables $x$ and $y$.

Forming and Solving the System of Equations

Equating the corresponding elements, we get the following system of four equations:

Element at position (1,1): $ 5x + 8 = 2 $
Element at position (1,2): $ 7 = 3y + 1 $
Element at position (2,1): $ y + 3 = 5 $
Element at position (2,2): $ 10x + 12 = 0 $

Now, let's solve these equations to find the values of $x$ and $y$.

Solving for x

Using equation 1:

$$ 5x + 8 = 2 $$ $$ 5x = 2 - 8 $$ $$ 5x = -6 $$ $$ x = \frac{-6}{5} $$

Using equation 4:

$$ 10x + 12 = 0 $$ $$ 10x = -12 $$ $$ x = \frac{-12}{10} $$ $$ x = \frac{-6}{5} $$

Both equations for $x$ give the consistent value $ x = \frac{-6}{5} $. This confirms our value for $x$.

Solving for y

Using equation 2:

$$ 7 = 3y + 1 $$ $$ 7 - 1 = 3y $$ $$ 6 = 3y $$ $$ y = \frac{6}{3} $$ $$ y = 2 $$

Using equation 3:

$$ y + 3 = 5 $$ $$ y = 5 - 3 $$ $$ y = 2 $$

Both equations for $y$ give the consistent value $ y = 2 $. This confirms our value for $y$.

Calculating the Value of 5x + 3y

We have found $ x = \frac{-6}{5} $ and $ y = 2 $. Now we need to find the value of the expression $ 5x + 3y $.

Substitute the values of $x$ and $y$ into the expression:

$$ 5x + 3y = 5\left(\frac{-6}{5}\right) + 3(2) $$ $$ 5x + 3y = -6 + 6 $$ $$ 5x + 3y = 0 $$

Therefore, the value of $ 5x + 3y $ is $ 0 $.

Revision Table: Key Concepts

Concept	Description	Application in Problem
Matrix Equality	Two matrices are equal if they have the same dimensions and their corresponding elements are equal.	Used to form a system of equations by equating elements.
Solving Linear Equations	Techniques to isolate variables in equations (e.g., addition, subtraction, multiplication, division).	Used to find the values of $x$ and $y$ from the system of equations.
Substitution	Replacing variables in an expression with their determined values.	Used to calculate the final value of $5x + 3y$.

Additional Information: Matrices and Systems of Equations

Matrix equality is a fundamental concept in linear algebra. It allows us to translate problems involving matrices into problems involving systems of linear equations, which we can then solve using various methods like substitution, elimination, or matrix methods (e.g., using inverse matrices or Gaussian elimination, although simple substitution was sufficient here).

A system of linear equations arises naturally in many scientific and engineering applications. The number of equations and the number of variables determine the nature of the solution (unique solution, infinite solutions, or no solution). In this case, we had four equations, but they provided consistent values for just two variables, confirming the matrices were indeed equal for specific $x$ and $y$.

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