Problem 50 - Algebra for College Students

Question

Evaluate the algebraic expressions in Problems 35-57 for the given values of the variables. \(3(x-2)-4(x+3), \quad x=-2\)

Step-by-Step Solution

Answer

The value of the expression is -16.

Step 1: Substitute the given value of x

The given expression is \(3(x-2)-4(x+3)\). We need to evaluate this when \(x = -2\). Substitute \(-2\) for \(x\) in the expression: \(3(-2-2)-4(-2+3)\).

Step 2: Simplify inside the parentheses

First, perform the operations inside the parentheses. Compute \((-2-2)\) and \((-2+3)\). This gives: \(3(-4)-4(1)\).

Step 3: Multiply by the coefficients

Now, multiply each of the results by their respective coefficients. Calculate \(3(-4)\) which equals \(-12\), and \(-4(1)\) which equals \(-4\).

Step 4: Combine the results

Combine the results from the multiplication step: \(-12 - 4\). This leads to \(-16\).

Key Concepts

Evaluating Expressions

Evaluating expressions in algebra involves finding the value of an expression for given variable values. It is like solving a math puzzle where each letter represents a number.

• To start, identify the parts of the expression. In our example, we have the expression \(3(x-2)-4(x+3)\).
• We need to evaluate it for \(x = -2\). This means we will replace every instance of \(x\) with \(-2\).
• Carefully perform each operation step by step to ensure accuracy.

Evaluating ensures you are following the correct sequence and applying operations properly, leading to the correct final answer.

Variable Substitution

Variable substitution is the process of replacing a variable with a given value to simplify and evaluate an expression.

• For our expression \(3(x-2)-4(x+3)\), replace \(x\) with \(-2\).
• The expression transforms to \(3(-2-2)-4(-2+3)\) after substitution.

Substitution is crucial because any misplacement can lead to incorrect answers. Always double-check your replacements.

Parentheses Simplification

After substituting the variables, focus on simplifying the expressions within parentheses first. This step adheres to the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right).

• For \(3(-2-2)-4(-2+3)\), simplify \((-2-2)\) and \((-2+3)\).
• This results in \(3(-4)-4(1)\).

Simplifying the parentheses first helps in reducing the complexity and making subsequent calculations simpler.

Arithmetic Operations

Once the parentheses are simplified, apply the necessary arithmetic operations based on their order in the expression.

• First, perform multiplication: \(3(-4)\) results in \(-12\), and \(-4(1)\) results in \(-4\).
• Then, combine these results: \(-12 - 4\).
• The answer is \(-16\).

Understanding arithmetic operations and their sequence ensures that each step leading to the final value is executed correctly. Always remember to take it step by step to avoid mistakes.