Problem 61 - Algebra 1: Concepts and Skills

Question

MULTIPLE CHOICE Which is a solution of the equation \(5(8-x)=25 ?\) A. 2 B. 3 C. 4 D. 5

Step-by-Step Solution

Answer

The correct answer is B. 3

Step 1: Distribute the multiplier

Begin by distributing the '5' inside the parentheses: \(5*8 - 5*x = 25\). This simplifies to \(40 - 5x = 25\).

Step 2: Isolate the 'x' variable

Subtract '40' from both sides of the equation to isolate the 'x' term: \(-5x = 25 - 40\), which simplifies to \(-5x = -15\).

Step 3: Solve for 'x'

Divide by '-5' on both sides of the equation to solve for 'x': \(x = -15 / -5\). The solution is \(x = 3\).

Key Concepts

Algebraic Expressions

Understanding algebraic expressions is fundamental in solving equations. An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. For example, in the equation given \(5(8-x)=25\), '\(5(8-x)\)' is an algebraic expression. The numbers in this expression, such as 5 and 8, are called coefficients and constants, respectively. The 'x' represents a variable, which is an unknown value we aim to find. When we combine these elements using mathematical operations such as addition, subtraction, multiplication, or division, we form an algebraic equation that can be solved for the variable.

To solve for 'x' in our example, algebraic manipulation is used - a step by step method that applies arithmetic operations to both sides of the equation, ultimately revealing the variable's value.

Isolating Variables

Isolating the variable in an algebraic equation is a critical step toward finding its solution. It involves rearranging the equation so that the variable we're solving for is on one side of the equation, and all other numbers or terms are on the opposite side. In the process of isolating variables, we do our best to unravel the equation, pulling the variable out of complex expressions.

In the equation \(5(8-x)=25\), we must perform operations that will move all other terms away from the 'x'. This approach often involves reversing the operations that are being applied to the variable. In this case, we would start by distributing the 5 across the \(8-x\) and then subtracting 40 from both sides. By doing so, we separate the variable from the rest of the equation, making it clear and ready to be solved.

Distributive Property

The distributive property is a useful algebraic rule that allows us to multiply a single term by each term inside a set of parentheses. The classic form of the distributive property is \(a(b + c) = ab + ac\). This rule is essential when dealing with algebraic expressions that include parentheses.

In the original problem \(5(8-x)=25\), we apply the distributive property by multiplying the number outside the parentheses (5) with each term inside the parentheses (8 and '-x'). This simplifies the equation to \(40 - 5x = 25\), setting the stage for the other steps needed in solving the equation. Understanding how to properly distribute ensures that we can simplify expressions correctly and advance toward isolating the variable to solve the equation.