Question

In each case, simplify the given expression, if possible. \(3 x-2 y+7 x-11 y\)

Short answer

Question: Simplify the following algebraic expression: \(3x-2y+7x-11y\). Answer: The simplified expression is \(10x - 13y\).

Step-by-step solution

Identify like terms

In the given expression, \(3x-2y+7x-11y\), we can identify two pairs of like terms: \(3x\) and \(7x\) (both x terms), and \(-2y\) and \(-11y\) (both y terms).

Combine like terms (x terms)

We can start by combining the x terms, \(3x\) and \(7x\). To do this, we add the coefficients (the numbers attached to the variables): \((3+7)x\). This results in \(10x\).

Combine like terms (y terms)

Next, we combine the y terms, \(-2y\) and \(-11y\). To do this, we add the coefficients (the numbers attached to the variables): \((-2+-11)y\). This results in \(-13y\).

Write the simplified expression

Now that we've combined both pairs of like terms, we can write our simplified algebraic expression by putting the results from Steps 2 and 3 together. This gives us: $$10x - 13y$$

Key concepts

Like Terms

In algebra, like terms are terms in an expression that have exactly the same variables raised to the same powers. For example, in the expression \(3x - 2y + 7x - 11y\), both terms \(3x\) and \(7x\) are like terms because they share the variable \(x\). Similarly, \(-2y\) and \(-11y\) are also like terms because they share the variable \(y\).
Recognizing like terms is crucial when simplifying algebraic expressions. This is because only like terms can be combined—unlike terms cannot be added or subtracted. So, when identifying like terms, always look for terms with identical variable parts.

Simplifying Expressions

Simplifying expressions involves transforming a mathematical expression into its most basic form while maintaining its original value. It's an essential process that makes expressions easier to interpret and use in equations.
When simplifying, your goal is to combine like terms and reduce the expression to avoid redundancy. Start by breaking down each part of the expression and look for opportunities to combine terms. This not only makes an expression neater but also often easier to work with, especially in more complex calculations.

• Identify the different terms within the expression.
• Group and combine like terms together.
• Perform arithmetic operations to simplify further.

For instance, in the expression \(3x - 2y + 7x - 11y\), we simplify by combining the \(x\) terms and the \(y\) terms, resulting in \(10x - 13y\).

Combining Like Terms

Combining like terms means summing or subtracting the coefficients of like terms while keeping the common variable part intact. This process is crucial for simplifying expressions.
Let's break it down with the example expression \(3x - 2y + 7x - 11y\):

• First, the \(x\) terms \(3x\) and \(7x\) are combined by adding their coefficients: \(3 + 7 = 10\). Thus, you get \(10x\).
• Next, the \(y\) terms \(-2y\) and \(-11y\) are combined by adding their coefficients: \(-2 + (-11) = -13\). This results in \(-13y\).

By efficiently combining like terms, you not only simplify the expression but also make it clearer. This makes further algebraic manipulations easier and reduces the likelihood of errors. The result \(10x - 13y\) is the simplified form of the original expression.