Problem 53 - Intermediate Algebra

Question

Evaluate the algebraic expressions for the given values of the variables. $$ 2(x-1)-(x+2)-3(2 x-1), \quad x=-1 $$

Step-by-Step Solution

Answer

The expression evaluates to 4.

Step 1: Substitute the Variable

Replace the variable \( x \) with the value given, which is \( x = -1 \). This results in the expression: \[ 2((-1)-1) - ((-1)+2) - 3(2(-1)-1). \]

Step 2: Simplify Within Parentheses

Start by simplifying each term inside the parentheses: - For \((x-1)\), substitute to get \((-1)-1 = -2\).- For \((x+2)\), substitute to get \((-1)+2 = 1\).- For \( (2x-1)\), substitute to get \( 2(-1) - 1 = -2 - 1 = -3\).The expression now becomes: \[ 2(-2) - (1) - 3(-3). \]

Step 3: Multiply Each Term by its Coefficient

Multiply each simplified term by its respective coefficient:- For \( 2(-2) \), multiply to get \(-4\).- For \( 1 \), it remains as \(-1\).- For \( 3(-3) \), multiply to get \(-9\).The expression is now \[ -4 - 1 + 9. \]

Step 4: Evaluate the Expression

Perform the addition and subtraction from left to right:- Start with \(-4 - 1 = -5\).- Then \(-5 + 9 = 4\).Thus, the simplified expression gives us the final result, \( 4 \).

Key Concepts

Substitution in Algebra

In algebra, substitution is a fundamental concept used to evaluate expressions. It involves replacing variables with their assigned values and solving. Imagine having to figure out the value of a mystery box; substitution is like being given the missing number to complete the puzzle.

To substitute correctly:

• Identify the variable in the algebraic expression. In our example, it's \( x \).
• Replace each instance of the variable with the number given. Here, we substitute \( x = -1 \).
• Rewrite the expression with numbers in place of variables. For instance, replace \( x \) in \( 2(x-1)-(x+2)-3(2x-1) \) with -1 to start the evaluation process.

Using substitution efficiently helps simplify complex problems, serving as the first step toward finding the solution.

Simplifying Expressions

After substituting, the next step in solving algebraic expressions is simplification. Simplifying involves reducing expressions to their most basic form, which makes them easier to solve.

During simplification:

• First, resolve the operations inside the parentheses. For example, \((-1)-1\) simplifies to \(-2\), and \(2(-1)-1\) simplifies to \(-3\).
• Once inside terms are simplified, multiply them by their respective coefficients. Here, \( 2(-2) \) becomes \(-4\) and \( 3(-3) \) yields \(-9\).

Simplification is crucial as it breaks down complicated expressions into smaller, manageable parts. This reduces potential errors in calculations and ultimately, guides you toward the solution. The goal here is to take the simplified terms and combine them through addition and subtraction.

Order of Operations

Understanding the order of operations is essential in algebra to ensure that expressions are evaluated correctly. The standard protocol is captured by the acronym PEMDAS—Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

Applying this sequence helps:

• Solve expressions inside parentheses first. This was seen when we simplified \( (x-1), (x+2), \) and \( (2x-1) \).
• Next, attend to multiplication or division in order. We multiply simplified terms by their coefficients at this stage.
• Finally, handle addition or subtraction from left to right. In our example, after all multiplication, we perform \(-4 - 1 + 9\), adhering to this order.

Keeping the order of operations in mind ensures a logical approach to solving fractions, preventing mistakes and yielding accurate results. Consistently applying PEMDAS can simplify even the most daunting algebraic expressions into straightforward solutions.