Problem 37
Question
In the following exercises, simplify the following expressions by combining like terms. $$ 7 x+2+3 x+4 $$
Step-by-Step Solution
Verified Answer
10x + 6
1Step 1: Identify Like Terms
Like terms are terms that contain the same variables raised to the same power. In the expression, the like terms are the terms containing the variable 'x' and the constant terms.
2Step 2: Group Like Terms Together
Group the terms containing 'x' together and the constant terms together. So, from the expression, we have: \( 7x + 3x \) and \( 2 + 4 \).
3Step 3: Combine Like Terms
Add the coefficients of the 'x' terms together and add the constant terms together: \( 7x + 3x = 10x \) and \( 2 + 4 = 6 \).
4Step 4: Write the Simplified Expression
Combine the results from the previous step into a single expression: \( 10x + 6 \).
Key Concepts
Combining Like TermsAlgebraic SimplificationCoefficients
Combining Like Terms
Combining like terms is crucial in simplifying algebraic expressions. 'Like terms' are terms that contain the same variables raised to the same power. For example, in the expression \(7x + 2 + 3x + 4\), the terms \(7x\) and \(3x\) are like terms because they both contain the variable 'x'. Similarly, the terms 2 and 4 are like terms because they are constants.
To combine like terms, you simply add or subtract their coefficients. In our example, you group \(7x\) and \(3x\) together and then add the coefficients: \(7 + 3 = 10\). For the constants, you group 2 and 4 together and add them: \(2 + 4 = 6\). Therefore, the simplified expression is \(10x + 6\). Combining like terms makes it easier to work with and understand algebraic expressions.
To combine like terms, you simply add or subtract their coefficients. In our example, you group \(7x\) and \(3x\) together and then add the coefficients: \(7 + 3 = 10\). For the constants, you group 2 and 4 together and add them: \(2 + 4 = 6\). Therefore, the simplified expression is \(10x + 6\). Combining like terms makes it easier to work with and understand algebraic expressions.
Algebraic Simplification
Algebraic simplification is the process of making an algebraic expression as straightforward as possible. By simplifying, you reduce the complexity of the expression without changing its value.
For instance, in the expression \(7x + 2 + 3x + 4\), you identify and combine like terms to simplify it. First, we notice the like terms: \(7x\) and \(3x\) as well as the constants 2 and 4.
Next, we combine the like terms: \(7x + 3x = 10x\) and \(2 + 4 = 6\).
Therefore, the simplified expression is \(10x + 6\). Simplifying algebraic expressions helps in solving equations and understanding the relationships between different variables more clearly. It is a fundamental skill in algebra that you'll use frequently.
For instance, in the expression \(7x + 2 + 3x + 4\), you identify and combine like terms to simplify it. First, we notice the like terms: \(7x\) and \(3x\) as well as the constants 2 and 4.
Next, we combine the like terms: \(7x + 3x = 10x\) and \(2 + 4 = 6\).
Therefore, the simplified expression is \(10x + 6\). Simplifying algebraic expressions helps in solving equations and understanding the relationships between different variables more clearly. It is a fundamental skill in algebra that you'll use frequently.
Coefficients
Coefficients are the numerical factors in terms of an algebraic expression. For instance, in the term \(7x\), the coefficient is 7. In \(3x\), the coefficient is 3.
Understanding coefficients is essential for combining like terms and simplifying expressions. When you combine like terms, you add or subtract their coefficients. In the expression \(7x + 3x\), you add the coefficients 7 and 3 to get 10, resulting in \(10x\).
Coefficients also help in understanding the magnitude of terms. Higher coefficients indicate larger contributions from those terms in the overall expression. For example, in \(7x + 3x\), the term \(7x\) contributes more than the term \(3x\), but when combined, they magnify the effect together. Recognizing and manipulating coefficients allows you to simplify and solve complex algebraic expressions efficiently.
Understanding coefficients is essential for combining like terms and simplifying expressions. When you combine like terms, you add or subtract their coefficients. In the expression \(7x + 3x\), you add the coefficients 7 and 3 to get 10, resulting in \(10x\).
Coefficients also help in understanding the magnitude of terms. Higher coefficients indicate larger contributions from those terms in the overall expression. For example, in \(7x + 3x\), the term \(7x\) contributes more than the term \(3x\), but when combined, they magnify the effect together. Recognizing and manipulating coefficients allows you to simplify and solve complex algebraic expressions efficiently.
Other exercises in this chapter
Problem 35
In the following exercises, evaluate the following expressions. When \(x=10, y=7\) \((x-y)^{2}\)
View solution Problem 36
In the following exercises, evaluate the following expressions. When \(a=3, b=8\) \(a^{2}+b^{2}\)
View solution Problem 38
In the following exercises, simplify the following expressions by combining like terms. $$ 8 y+5+2 y-4 $$
View solution Problem 39
In the following exercises, simplify the following expressions by combining like terms. $$ 10 a+7+5 a-2+7 a-4 $$
View solution