Problem 77 - Elementary Algebra

Question

Simplify. $$ \left(2 x_{2}-7 x+1\right)+(x 2+7 x-5) $$

Step-by-Step Solution

Answer

The expression simplifies to \(3x_2 - 4\).

Step 1: Identify Like Terms

In the expression \((2x_2 - 7x + 1) + (x2 + 7x - 5)\), identify like terms. Like terms are terms with the same variables raised to the same power. Here, we have two sets of like terms: \(2x_2\) and \(x2\) (which seems like a notation error and is the same as \(x_2\)), \(-7x\) and \(7x\), and the constant terms \(1\) and \(-5\).

Step 2: Simplify Like Terms Involving x_2

Add the coefficients of the terms involving \(x_2\), which are \(2x_2\) and \(x_2\). This results in \(2x_2 + x_2 = 3x_2\).

Step 3: Simplify Like Terms Involving x

The terms \(-7x\) and \(7x\) cancel each other out when added, resulting in \(-7x + 7x = 0x\), which means there is no \(x\) term in the final simplified expression.

Step 4: Simplify Constant Terms

Add the constant terms. We have \(1\) and \(-5\), which results in \(1 + (-5) = -4\).

Step 5: Final Simplified Expression

Combine the results from Steps 2 to 4 to obtain the final simplified expression: \(3x_2 - 4\).

Key Concepts

Like Terms

In algebra, like terms are a fundamental concept that makes simplifying expressions possible. Like terms are terms that have the exact same variable parts, meaning the variables and their exponents are identical. This similarity allows us to combine them using basic arithmetic operations. For instance, in the expression

• \(2x_2\)
• \(x_2\)

Both of these are like terms because they involve the same variable \(x_2\). Similarly, when we encounter terms like \(-7x\) and \(7x\), they too are like terms as they share the variable \(x\). Recognizing like terms is the first step in simplifying algebraic expressions because it tells us which terms we can add or subtract.

Simplifying Expressions

Simplifying an algebraic expression involves reducing it to its simplest form by performing all possible arithmetic operations and combining like terms. This process makes expressions shorter and easier to handle. When simplifying expressions, we

• Group like terms together. For example, in the expression \((2x_2 - 7x + 1) + (x_2 + 7x - 5)\), we group terms with \(x_2\), \(x\), and the constants separately.
• Perform arithmetic operations on the coefficients of the like terms. This means we add or subtract the numbers in front of the variables rather than the variables themselves.

By simplifying, we make complex algebraic expressions manageable and find solutions more efficiently.

Combining Like Terms

Once like terms are identified, they can be combined, leading to a simpler expression. Combining like terms involves adding or subtracting their coefficients while keeping the variable part unchanged. This is a straightforward yet essential step. Let's consider our previous example. We identified the like terms:

• \(2x_2\) and \(x_2\), which combine to form \(3x_2\) because \(2 + 1 = 3\)
• \(-7x\) and \(7x\), which cancel each other out, resulting in \(0\)
• The constants \(1\) and \(-5\), which add up to \(-4\)

Thus, combining like terms effectively reduces the expression \((2x_2 - 7x + 1) + (x_2 + 7x - 5)\) down to \(3x_2 - 4\). This step is crucial for solving equations and inequalities as it streamlines the problem-solving process.