Find the values of the variables for which each statement is true, if possible. [x+2y−6z−3w+5]=[−2803]\left[ \begin{matrix} x + 2 & y - 6 \\ z - 3 & w + 5 \end{matrix} \right] = \left[ \begin{matrix} -2 & 8 \\ 0 & 3 \end{matrix} \right]

Ch. 5 - Systems and Matrices

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Lial 13th Edition

Ch. 5 - Systems and Matrices

문제 15

6장, 문제 15

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

검증된 단계별 안내

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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

Find the values of the variables for which each statement is true, if possible. [x+2y−6z−3w+5]=[−2803]\(\left\)[ \(\begin{matrix}\) x + 2 & y - 6 \\ z - 3 & w + 5 \(\end{matrix}\) \(\right\)] = \(\left\)[ \(\begin{matrix}\) -2 & 8 \\ 0 & 3 \(\end{matrix}\) \(\right\)]