Chapter 11: Problem 5
Short Answer
Expert verified
The ordered pair makes .
Step by step solution
01
- Understand the problem
Given a function of two variables, , determine for which of the provided ordered pairs , the value of is not equal to zero.
02
- Substitute pair (-3, 2)
Substitute and into the equation. Since , the pair makes .
03
- Substitute pair (-2, 3)
Substitute and into the equation. Since , the pair does not make .
04
- Substitute pair
Substitute and into the equation. Since , the pair does not make .
05
- Substitute pair
Substitute and into the equation. Since , the pair does not make .
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Variable Substitution
Variable substitution is a crucial technique in solving mathematical problems, especially in algebra. This concept involves replacing variables in an equation with given numerical values.
For example, in the given problem, we have the function of two variables: . To check which ordered pair results in , we substitute the values of and from each pair into the equation:
results in a non-zero value.
For example, in the given problem, we have the function of two variables:
- For the pair
, we substitute and into the equation. - For the pair
, substitute and . - For the pair
, substitute and . - For the pair
, substitute and .
Quadratic Equations
Quadratic equations are algebraic expressions where the highest power of the variable is squared. In this problem, the term indicates that we are dealing with a quadratic equation in one of the variables.
When substituting values, you can notice how the squared term affects the overall value of :
Formula example with substitution:
For , the term becomes . This significant value has a large impact on .
Due to the squared term, small changes in can cause large changes in , highlighting the importance of careful substitution and simplification when dealing with quadratic components of an equation.
When substituting values, you can notice how the squared term affects the overall value of
Formula example with substitution:
For
Due to the squared term, small changes in
Algebraic Manipulations
Algebraic manipulations refer to the processes of rearranging and simplifying expressions. In our problem, after substituting the values into the equation, we perform several steps to simplify the expressions:
This manipulation shows how each component of the equation contributes to the final value of .
- Expand any squared terms, like
. - Multiply constants and variables, such as
and . - Combine like terms, which means adding or subtracting terms with the same variables.
This manipulation shows how each component of the equation contributes to the final value of
Problem-Solving Techniques
Effective problem-solving techniques involve understanding the problem, systematic substitution, careful simplification, and evaluation of results. Here's how you can approach this type of problem:
- First, interpret the problem to understand what is required – in this case, finding when
. - Next, substitute the given pairs
into the function one at a time. - Simplify the resulting equations through careful algebraic manipulations and combine like terms.
- Finally, evaluate whether the simplified result is zero or not, confirming if the pair makes
.