Problem 52 - Algebra for College Students

Question

Evaluate the algebraic expressions in Problems 35-57 for the given values of the variables. \(-4(2 x-1)+7(3 x+4), \quad x=4\)

Step-by-Step Solution

Answer

The evaluated expression is 84.

Step 1: Substitute the Given Value

First, substitute \( x = 4 \) into the expression. The algebraic expression is \(-4(2x - 1) + 7(3x + 4)\). By substituting, it becomes \(-4(2(4) - 1) + 7(3(4) + 4)\).

Step 2: Simplify Inside the Parentheses

Calculate the values inside the parentheses:- \(2(4) = 8\), so \(2(4) - 1 = 8 - 1 = 7\).- \(3(4) = 12\), so \(3(4) + 4 = 12 + 4 = 16\).

Step 3: Multiply Coefficients with Simplified Values

Calculate the multiplication:- \(-4 \cdot 7 = -28\).- \(7 \cdot 16 = 112\).

Step 4: Add the Products

Add the results from Step 3:\(-28 + 112 = 84\).

Key Concepts

Substitution in Algebra

Substitution in algebra involves replacing the variable in an expression with a given number or value. It is an essential step in evaluating algebraic expressions and makes them easier to work with. Think of it as "plugging in" values to see what happens to the expression. For example, if we have a variable \( x \) and we're asked to evaluate an expression for \( x = 4 \), we replace every instance of \( x \) with 4 in the expression.For example, with the expression \(-4(2x - 1) + 7(3x + 4)\), we substitute \( x = 4 \), leading to {-4(2(4) - 1) + 7(3(4) + 4)}. Substitution helps us transition from algebraic scenarios to numerical calculations, paving the way for further simplifications and final solutions.

Simplifying Expressions

After substituting values, the next step in evaluating algebraic expressions is simplifying. This involves reducing the expression to its simplest form by performing arithmetic operations. In our exercise, once we substitute \( x = 4 \), we simplify:

• Inside the parentheses, start by handling multiplication because it precedes subtraction and addition. Hence, \(2(4) = 8\) so \(2(4) - 1 = 8 - 1 = 7\).
• Similarly, \(3(4) = 12\) gives \(3(4) + 4 = 12 + 4 = 16\).

Simplifying removes complexity by breaking down expressions into more manageable calculations. By the end of this process, your expression should be free of operations inside the parentheses, leaving a clear path to final calculation.

Order of Operations

The order of operations is a fundamental principle in mathematics that determines the sequence in which calculations are performed to achieve the correct result. The acronym PEMDAS is often used to remember the order: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).In the given expression, once values are substituted and expressions inside parentheses simplified, follow these steps:

• Perform necessary multiplications \(-4 \cdot 7 = -28\) and \(7 \cdot 16 = 112\). Multiplication and division come next after handling the parentheses.
• Finally, perform the addition or subtraction, where you combine -28 and 112 to get 84.

Following the order of operations ensures that everyone gets the same, correct result for the same expression.