Fill in the blank(s) to correctly complete each sentence. For the following statement to be true, the value of x must be ____, and the value of y must be ____. [ 3 − 6 5 1 ] = [ x + 1 − 6 5 y + 1 ] \left[ \begin{matrix} 3 & -6 \\ 5 & 1 \end{matrix} \right] = \left[ \begin{matrix} x + 1 & -6 \\ 5 & y + 1 \end{matrix} \right]

Hello, everyone. We are asked to evaluate P and Q for the following statement to be true. The statement we are given has a two by two matrix where the first row is negative 57. The second row is 21 and that is equal to another two by two matrix where the first row has the elements P minus seven and seven. The second row has the elements two and Q minus three. Our answer choices are A P equals four, Q equals negative two B P equals negative two, Q equals four, C P equals four, Q equals two D P equals two, Q equals four. Since both matrices are two by two, and I can see that the elements that do not have a variable match, it is safe to assume that the corresponding locations in the matrix are equal to each other. So we can set negative five, equal to P minus seven. And we will also set one equal to Q minus three. And then we'll solve for both of these variables. First solving for P again, I have negative five equals P minus seven. So I add seven to both sides and I get that P equals two, next one equals Q minus three, gonna add three to both sides. And I get that four equals Q or Q equals four. And those are our answers. That would be what makes this statement true. And it's answer choice D P equals two, Q equals four. 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

2:05 minutes

Problem 1

Textbook Question

Fill in the blank(s) to correctly complete each sentence. For the following statement to be true, the value of x must be , and the value of y must be .
$[\begin{matrix} 3 & - 6 \\ 5 & 1 \end{matrix}] = [\begin{matrix} x + 1 & - 6 \\ 5 & y + 1 \end{matrix}]$

Verified step by step guidance

First, identify the complete statement or equation that relates the variables x and y. Without the full equation or context, we cannot determine the values of x and y.

Once the equation or condition is known, isolate one variable in terms of the other if possible. For example, if the equation is of the form $Ax + By = C$, solve for $y$ as $y = \frac{C - Ax}{B}$ or for $x$ as $x = \frac{C - By}{A}$.

Determine any restrictions or conditions on the variables, such as domain restrictions or values that make the equation true. This might involve solving for specific values or ranges.

Substitute any known values or conditions back into the equation to find the corresponding values of the other variable.

Verify the solution by plugging the values of $x$ and $y$ back into the original equation or statement to ensure it holds true.

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Key Concepts

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

Understanding Variables and Their Domains

Variables like x and y represent unknown values that can vary within certain constraints called domains. Knowing the domain helps determine which values x and y can take to satisfy an equation or inequality.

Interpreting Equations and Statements

To complete sentences involving variables, one must understand the given equation or statement fully. This involves identifying what conditions make the statement true and how the variables relate to each other.

Solving for Variables

Solving for variables means finding specific values or sets of values that satisfy an equation or inequality. This often involves algebraic manipulation such as isolating variables, factoring, or using substitution.

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