Problem 50 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The expression evaluates to \(-16\).

Step 1: Substitute the Variable

To solve the expression \(3(x-2) - 4(x+3)\), substitute \(x = -2\) into the expression. This gives: \(3((-2)-2) - 4((-2)+3)\).

Step 2: Simplify Inside the Parentheses

Simplify the expressions inside the parentheses: \((-2)-2 = -4\) and \((-2)+3 = 1\). The expression becomes \(3(-4) - 4(1)\).

Step 3: Multiply the Coefficients

Multiply the numbers: \(3(-4) = -12\) and \(-4(1) = -4\). So, the expression now is \(-12 - 4\).

Step 4: Perform the Subtraction

Subtract \(-12 - 4\) to get \(-16\). Thus, the expression evaluates to \(-16\).

Key Concepts

Substitution Method

The substitution method is a critical skill for evaluating algebraic expressions. This method involves replacing a variable in the expression with its given value. In our exercise, the expression is \(3(x-2)-4(x+3)\), and we need to evaluate it for \(x = -2\).
By substituting \(x = -2\) into the expression, we get the new form: \(3((-2)-2) - 4((-2)+3)\).
The substitution method helps to "translate" the algebraic expression into numerical form, making the problem easier to solve as it converts the variable-based equation into simple numerical operations.

Simplifying Expressions

Simplifying expressions is about reducing them to their simplest forms. This can involve combining like terms or performing operations inside parentheses. In the provided example, once we substitute \(x = -2\) into the equation, we work on simplifying the inside of each parentheses.
- For the sub-expression \((-2)-2\), simplify to \(-4\).- For the sub-expression \((-2)+3\), simplify it to \(1\).Now, the original expression \(3(x-2)-4(x+3)\) becomes \(3(-4) - 4(1)\).
Simplifying helps reduce clutter, making it easier to perform further arithmetic operations and ultimately solve the problem.

Arithmetic Operations

Arithmetic operations are the fundamental math steps you take after substitution and simplification. These operations include addition, subtraction, multiplication, and division. In our example, we're initially left with a simplified expression: \(3(-4) - 4(1)\).
- **Multiplication:** First, multiply the coefficients of the terms: - \(3 \times -4 = -12\) - \(-4 \times 1 = -4\)- **Subtraction:** After the multiplication, we subtract the results: - Combine to get \(-12 - 4\), which results in \(-16\).
Mastering these operations helps solve expressions efficiently and is essential for handling more complex math problems in algebra.