Problem 56 - Elementary Algebra

Question

Simplify. $$ 1 x 2-x-6+1 x 2-3 x-10 $$

Step-by-Step Solution

Answer

The simplified expression is \(-4x - 12\).

Step 1: Recognize Terms

Identify the like terms in the expression. The expression is: \[1 \times 2 - x - 6 + 1 \times 2 - 3x - 10\]Realize that \(1 \times 2\) is just 2, simplifying to: \[2 - x - 6 + 2 - 3x - 10\]

Step 2: Combine Like Constants

Add together the constant numbers: Identify and combine all the numbers without variables:\[2 - 6 + 2 - 10\]Calculate:\[2 - 6 = -4\]\[-4 + 2 = -2\]\[-2 - 10 = -12\]Therefore, the expression now is:\[- x - 12 - 3x\]

Step 3: Combine Like Variable Terms

Combine the like terms with the variable \(x\):\[- x - 3x\]This can be simplified by adding the coefficients of \(x\):\[-1x - 3x = -4x\]Thus, the expression becomes:\[-4x - 12\]

Step 4: Final Simplified Form

Write the final simplified version of the expression:The simplified expression is:\[-4x - 12\]

Key Concepts

Combining Like Terms

In algebra, simplifying expressions often involves combining like terms. Like terms are those that have the exact same variables raised to the same powers. The process to simplify by combining like terms helps to reduce the complexity of an expression, making it easier to work with.
To combine like terms, follow these steps:

• Identify terms with identical variables and exponents.
• Sum or subtract the coefficients of these terms.

For instance, consider terms like \(-x - 3x\). Both terms involve the variable \(x\) and thus are like terms. Here, you would add the coefficients: \(-1\) and \(-3\), to obtain \(-4x\).
By understanding which terms can be combined, you can simplify expressions effectively, bringing clarity to complex problems.

Coefficients in Algebra

Coefficients are a key element in algebraic expressions, representing the numerical part of terms with variables. In the term \(-3x\), the coefficient is \(-3\). Coefficients indicate how many times to multiply the variable they accompany.
When simplifying expressions, the focus is on the coefficients of like terms:

• Combine them by performing operations such as addition or subtraction.
• Keep the variable part unchanged and consistent.

Take the expression \(-x - 3x\) as an example; here, the coefficients \(-1\) and \(-3\) must be summed to form \(-4x\). Understanding coefficients allows you to effectively manage and transform expressions.

Constants in Algebraic Expressions

Constants are numbers in algebraic expressions that are not multiplied by any variables. They are the standalone figures in the expression, such as \(2\), \(-6\), and \(-10\) in this exercise.
To simplify expressions involving constants, you add or subtract them to reduce the expression to its simplest form.

• Identify all constants present in the expression.
• Combine them together using addition or subtraction.

For our expression, you take \(2 - 6 + 2 - 10\) and combine them through straightforward arithmetic steps to arrive at \(-12\). Recognizing and managing constants is a vital skill as it provides the final touch in simplifying an algebraic expression.