Chapter 5: Problem 10
In each case, simplify the given expression, if possible. \(7 \alpha-3 \beta+2 \gamma-7 \alpha+11 \beta\)
Short Answer
Expert verified
Question: Simplify the expression \(7\alpha - 3\beta + 2\gamma - 7\alpha + 11\beta\).
Answer: \(8\beta + 2\gamma\)
Step by step solution
01
Identify like terms
In the expression, we need to identify the terms with the same variables. The like terms are:
- \(7\alpha\) and \(-7\alpha\)
- \(-3\beta\) and \(11\beta\)
- \(2\gamma\)
02
Combine like terms
Now that we have identified the like terms, we need to combine them. To combine the like terms with the same variable, we will add or subtract the coefficients of the terms.
For the α terms:
\(7\alpha - 7\alpha = 0\alpha\)
For the β terms:
\(-3\beta + 11\beta = 8\beta\)
For the γ terms, there is only one term:
\(2\gamma\)
03
Write the simplified expression
Now that we have combined the like terms, we can write the simplified expression. Since \(0\alpha\) is equal to 0, it will not be included in the simplified expression. Therefore, the simplified expression will be:
\(8\beta + 2\gamma\)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Combining Like Terms
When simplifying algebraic expressions, one of the most fundamental techniques is combining like terms. Like terms are terms within an expression that share the same variable and exponent. For instance, in the exercise given as \(7 \alpha - 3 \beta + 2 \gamma - 7 \alpha + 11 \beta\), the terms \(7 \alpha\) and \(-7 \alpha\) are like terms because they both contain the variable \(\alpha\), likewise, for the terms with \(\beta\).
To effectively combine like terms, you simply add or subtract their coefficients. A coefficient is a numerical value that is placed in front of a variable. Examples from the exercise demonstrate how \(7 \alpha\) added to \(-7 \alpha\) cancels out as the coefficients \(+7\) and \(-7\) sum to zero. Similarly, \(-3 \beta\) and \(+11 \beta\) result in \(+8 \beta\) when combined.
This process of combination helps to streamline expressions, making them simpler and easier to understand or further manipulate. Remember that terms with different variables or exponents are not like terms and cannot be combined in this manner. Always look for terms that match exactly in variable composition to successfully combine them.
To effectively combine like terms, you simply add or subtract their coefficients. A coefficient is a numerical value that is placed in front of a variable. Examples from the exercise demonstrate how \(7 \alpha\) added to \(-7 \alpha\) cancels out as the coefficients \(+7\) and \(-7\) sum to zero. Similarly, \(-3 \beta\) and \(+11 \beta\) result in \(+8 \beta\) when combined.
This process of combination helps to streamline expressions, making them simpler and easier to understand or further manipulate. Remember that terms with different variables or exponents are not like terms and cannot be combined in this manner. Always look for terms that match exactly in variable composition to successfully combine them.
Algebraic Simplification
Algebraic simplification is a process that aims to rewrite expressions in their simplest form. It includes a variety of techniques, and combining like terms is an initial step. Beyond combining like terms, simplification may involve expanding expressions, factoring, cancelling common factors in fractions, and applying the distributive property where necessary.
An important stage in algebraic simplification is to eliminate unnecessary elements. For instance, any term with a coefficient of zero, such as \(0\alpha\), becomes null and can be removed from the expression. This elimination is evident in our starting exercise where after combining like terms, \(7 \alpha - 7 \alpha\) becomes \(0\alpha\) and thus is omitted from the final simplified expression, showing as \(8 \beta + 2 \gamma\).
Each stage of simplification gets you closer to an expression that is easier to evaluate or manipulate further in equations and functions. As a strategy, always aim to reduce expressions to the least number of terms possible without altering the mathematical meaning, ensuring clarity and conciseness.
An important stage in algebraic simplification is to eliminate unnecessary elements. For instance, any term with a coefficient of zero, such as \(0\alpha\), becomes null and can be removed from the expression. This elimination is evident in our starting exercise where after combining like terms, \(7 \alpha - 7 \alpha\) becomes \(0\alpha\) and thus is omitted from the final simplified expression, showing as \(8 \beta + 2 \gamma\).
Each stage of simplification gets you closer to an expression that is easier to evaluate or manipulate further in equations and functions. As a strategy, always aim to reduce expressions to the least number of terms possible without altering the mathematical meaning, ensuring clarity and conciseness.
Variable Term Manipulation
Variable term manipulation is the skillful handling of the parts of algebraic expressions that contain variables. This not only refers to combining like terms but also applies to how we add, subtract, multiply or divide these terms. With practice, one becomes proficient in recognizing patterns and understanding the behaviors of different algebraic operations as they apply to variable terms.
In the exercise example, manipulation was straightforward since we only dealt with addition and subtraction. However, complexities increase when multiplication and division enter the equation, especially when you encounter terms involving exponents or coefficients. In these cases, remember that the laws of exponents and distributive property can guide you in appropriately manipulating the terms.
In the exercise example, manipulation was straightforward since we only dealt with addition and subtraction. However, complexities increase when multiplication and division enter the equation, especially when you encounter terms involving exponents or coefficients. In these cases, remember that the laws of exponents and distributive property can guide you in appropriately manipulating the terms.