Problem 15 - Introductory and Intermediate Algebra for College Students 4th

Question

Let \(x\) represent the number. Use the given conditions to write an equation. Solve the equation and find the number. Twice the sum of four and a number is \(36 .\) Find the number.

Step-by-Step Solution

Answer

The number is 14.

Step 1: Translate the problem to an equation

From the problem 'Twice the sum of four and a number is 36', the sentence can be translated into an equation. 'A number' can be represented as \(x\), 'sum of four and a number' is translated to \(4 + x\) and 'twice the sum' is \(2(4 + x)\). Thus, the entire problem can be represented as: \(2(4 + x) = 36\)

Step 2: Simplify the equation

We can simplify this equation by distributing 2 across the terms inside the parentheses: \(2*4 + 2*x = 36\), which simplifies to \(8 + 2x = 36\).

Step 3: Solve the equation for \(x\)

To isolate \(x\), first subtract 8 from both sides of the equation to get \(2x = 28\). Then, divide both sides by 2 in order to solve for \(x\), which gives \(x = 14\).

Key Concepts

Translating Word Problems

When you come across word problems, the challenge is often not just about finding the solution but understanding what the problem is asking in mathematical terms. Translating a word problem into an algebraic equation is a critical skill that involves recognizing keywords and phrases that point towards certain mathematical operations.

For example, in the phrase "Twice the sum of four and a number is 36," we look for clues on how to construct our equation. "Twice" signals multiplication, "sum" suggests addition, and "is" generally denotes the equal sign.

• "A number" is represented by a variable, like \(x\).
• "The sum of four and a number" is expressed as \(4 + x\).
• "Twice the sum" translates to multiplying the sum by 2, resulting in \(2(4 + x)\).

The phrase "is 36" tells us that everything before equals 36, giving us the full equation \(2(4 + x) = 36\). Mastering this translation process is key to tackling any word problem effectively.

Solving Linear Equations

Solving linear equations is a fundamental skill in algebra that involves finding the value of the variable that makes the equation true. Once we've translated our word problem into the equation \(2(4 + x) = 36\), the next step is to solve for \(x\).

A strategic approach includes performing operations that simplify the equation step by step:

• First, address any parenthesis through distribution, resulting in the equation \(8 + 2x = 36\).
• Then, look to isolate the variable term by getting rid of constants on the same side as \(x\). Subtract 8 from both sides to obtain \(2x = 28\).
• Finally, solve for \(x\) by dividing each side by the coefficient of \(x\) (2 in this case). This leaves us with \(x = 14\).

By following these steps methodically, you'll be able to solve any linear equation with confidence.

Distributive Property

The distributive property is a valuable tool in algebra that allows us to rewrite expressions involving parentheses. It states that \(a(b + c) = ab + ac\). This principle is particularly handy when an equation includes a term that multiplies over a sum or difference.

In the original equation \(2(4 + x) = 36\), applying the distributive property involves spreading \(2\) across both terms inside the parentheses, transforming it into \(2 \cdot 4 + 2 \cdot x\) which simplifies to \(8 + 2x\).

• This step lets us simplify complex expressions and more easily work towards solving for \(x\).
• Understanding and utilizing the distributive property helps to break down and manage larger expressions efficiently.

By mastering the use of the distributive property, you can navigate and simplify even the most daunting algebraic equations.