Problem 58
Question
SIMPLIFYING EXPRESSIONS Simplify the expression by combining like terms. $$ 3 x^{2}+2 x^{2}-7 $$
Step-by-Step Solution
Verified Answer
The simplified expression is \(5x^2 - 7\).
1Step 1: Identify Like Terms
In the given expression \(3x^2 + 2x^2 -7\), the like terms are \(3x^2\) and \(2x^2\). Both terms have the same variable \(x\) raised to the power of \(2\).
2Step 2: Combine Like Terms
Combine the like terms by adding the coefficients. The coefficient of \(3x^2\) is \(3\) and the coefficient of \(2x^2\) is \(2\). Therefore, adding these gives \(3 + 2 = 5\). So, the combined term is \(5x^2\).
3Step 3: Write Down the Simplified Expression
Finally, write down the simplified expression. After combining the two like terms \(3x^2\) and \(2x^2\) into \(5x^2\), the simplified expression becomes \(5x^2 - 7\).
Key Concepts
Combining Like TermsAlgebraic CoefficientsSimplifying Expressions Step by Step
Combining Like Terms
When simplifying algebraic expressions, one of the key steps is to combine like terms. But what are 'like terms'? In an algebraic expression, like terms are terms that have the same variables raised to the same power. For instance, in the expression \(3x^2 + 2x^2 - 7\), \(3x^2\) and \(2x^2\) are like terms because they have the same variable \(x\) raised to the same exponent, which is 2.
To combine like terms, simply add or subtract the coefficients (the numbers in front of the variables) while keeping the variable part unchanged. In our example, we add the coefficients 3 and 2 which gives us 5, so combining \(3x^2\) and \(2x^2\) results in \(5x^2\). Remember, unlike terms cannot be combined in this way. For instance, \(x^2\) and \(x\) are not like terms and cannot be combined because the variables are raised to different powers.
Using bullet points for clarity:
To combine like terms, simply add or subtract the coefficients (the numbers in front of the variables) while keeping the variable part unchanged. In our example, we add the coefficients 3 and 2 which gives us 5, so combining \(3x^2\) and \(2x^2\) results in \(5x^2\). Remember, unlike terms cannot be combined in this way. For instance, \(x^2\) and \(x\) are not like terms and cannot be combined because the variables are raised to different powers.
Using bullet points for clarity:
- Identify like terms by matching variables and their exponents.
- Add or subtract their coefficients.
- Keep the variable part of the term unchanged.
Algebraic Coefficients
Algebraic coefficients are the numerical parts of terms in an algebraic expression that are multiplied by the variables. Understanding coefficients is crucial when it comes to simplifying expressions, as they are what you manipulate when combining like terms. In the expression \(3x^2 + 2x^2 - 7\), the numbers 3 and 2 are coefficients of the \(x^2\) terms.
Co-efficients tell us how many 'units' of the variable we have. If you have a coefficient of 3 for \(x^2\), it means you have three units of \(x^2\). When simplifying, you combine these units from like terms to find out how many units you have in total. In our example, combining the coefficients 3 and 2 gives us 5, so the simplified expression has five units of \(x^2\), expressed as \(5x^2\).
Important points to remember about coefficients include:
Co-efficients tell us how many 'units' of the variable we have. If you have a coefficient of 3 for \(x^2\), it means you have three units of \(x^2\). When simplifying, you combine these units from like terms to find out how many units you have in total. In our example, combining the coefficients 3 and 2 gives us 5, so the simplified expression has five units of \(x^2\), expressed as \(5x^2\).
Important points to remember about coefficients include:
- They are the numbers that multiply the variables in terms.
- They are added or subtracted when combining like terms.
- They represent the quantity of the variable you have.
Simplifying Expressions Step by Step
The process of simplifying expressions involves clear and methodical steps. First, identify like terms within the expression, as we did with \(3x^2 + 2x^2 - 7\), finding that \(3x^2\) and \(2x^2\) are like terms.
Second, you combine these like terms by adding or subtracting their coefficients. In the given example, we added 3 and 2 to get 5. This step is critical because it reduces the expression to fewer terms, making it easier to understand and work with.
Finally, you write the simplified expression. After combining like terms, you have a new expression that represents the same quantity in a simpler form. In our example, the expression simplifies to \(5x^2 - 7\), which is easier to interpret and use in subsequent calculations.
To review, the steps are:
Second, you combine these like terms by adding or subtracting their coefficients. In the given example, we added 3 and 2 to get 5. This step is critical because it reduces the expression to fewer terms, making it easier to understand and work with.
Finally, you write the simplified expression. After combining like terms, you have a new expression that represents the same quantity in a simpler form. In our example, the expression simplifies to \(5x^2 - 7\), which is easier to interpret and use in subsequent calculations.
To review, the steps are:
- Identify like terms.
- Combine like terms by manipulating coefficients.
- Write the simplified expression.
Other exercises in this chapter
Problem 57
Use a calculator to evaluate the expression. Round your answer to two decimal places. $$x^{2}+x-27.2 \text { when } x=-7$$
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Evaluate the expression for the given value(s) of the variable(s). $$\frac{3 a-4 b}{a b} \text { when } a=-\frac{1}{3} \text { and } b=\frac{1}{4}$$
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Find the terms of the expression. $$-3 x+5-8 y$$
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Decide whether the statement is true or false . If it is false, give a counterexample. $$(-a) \cdot(-b)=(-b) \cdot(-a)$$
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