Find the values of the variables for which each statement is true, if possible. [ x + 2 y − 6 z − 3 w + 5 ] = [ − 2 8 0 3 ] \left[ \begin{matrix} x + 2 & y - 6 \\ z - 3 & w + 5 \end{matrix} \right] = \left[ \begin{matrix} -2 & 8 \\ 0 & 3 \end{matrix} \right]

Hello, everyone. We are asked to determine if possible the values of the variables that satisfy the given statement, we are given two matrices. They are both two by two matrices. The first one reads the top row as J minus three K plus three, second row, L minus nine M plus 11. And that's equal to the matrix with the first row of four negative 10. In the second row of negative 18 30 we have four answer choices all of which have values for the variables J K L and M in slightly different orders. To find the values of these variables, we are going to match up the corresponding locations in our matrices. So they're both two by two, we can say they correspond directly. So the top left corner. So the first element in the first row of each matrix will be equal. So that would be J minus three equals four. So to solve for J I add three to both sides, and I know that the value of J is seven. Next, I'm gonna go to the right and match the second element of the first row of both matrices. So the first one would be K plus three and that equals negative 10. So again, I saw for Kay and when I subtract three from both sides, K equals negative 13, two variables down two to go, the first element of the second row will match and that will be L minus nine equal to negative 18. So I add nine to both sides. L must equal negative nine. And lastly the last or the second element in the second row of each matrix. So M plus 11 must equal 30. Subtracting 11 from both sides. M 19. So we have J equal to seven K equal to negative 13. L equal to negative nine M equal to 19. And that matches answer choice A have a nice day.

7. Systems of Equations & Matrices

Determinants and Cramer's Rule

College Algebra

7. Systems of Equations & Matrices

Determinants and Cramer's Rule

2:44 minutes

Problem 15

Textbook Question

Find the values of the variables for which each statement is true, if possible.
$[\begin{matrix} x + 2 & y - 6 \\ z - 3 & w + 5 \end{matrix}] = [\begin{matrix} - 2 & 8 \\ 0 & 3 \end{matrix}]$

Verified step by step guidance

First, carefully read the problem statement to identify the given equations or expressions involving the variables. Since the problem references Examples 1 and 2, review those examples to understand the type of equations or statements you are dealing with.

Next, write down the equations or inequalities explicitly. For example, if the problem involves solving for variables in an equation like $a + b = c$, write it clearly to analyze the relationships.

Then, isolate one variable in terms of the others if possible. For instance, solve for $a$ in terms of $b$ and $c$ by rearranging the equation: $a = c - b$.

After that, substitute the expression from the previous step into any other equations or conditions given to reduce the number of variables and find possible values.

Finally, check the solutions obtained by substituting back into the original statements to verify if they satisfy all conditions. If no values satisfy the statements, conclude that no solution exists.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Solving Equations

Solving equations involves finding the values of variables that make the equation true. This process may include isolating the variable using inverse operations such as addition, subtraction, multiplication, division, or applying algebraic properties.

Checking for Extraneous Solutions

When solving equations, especially those involving variables on both sides or radicals, some solutions may not satisfy the original equation. It is important to substitute found values back into the original equation to verify their validity.

Understanding Variable Constraints

Variables may have restrictions based on the context or the equation type, such as denominators not being zero or expressions under square roots being non-negative. Recognizing these constraints helps determine the domain and valid solutions.

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