Problem 20 - Algebra 1: Concepts and Skills

Question

Match the sentence with its equation. Let x represent the number. A number increased by 2 is 4. A. \(x-4=2\) B. \(x+2=4\) C. \(\frac{x}{4}=2\) D. \(2 x=4\)

Step-by-Step Solution

Answer

The correct match for the sentence 'A number increased by 2 is 4' is option B.

Step 1: Understand the Sentence

The sentence is 'A number increased by 2 is 4'. This implies that when 2 is added to a certain number, the result or the sum is 4. The number that needs to be found is represented by \(x\).

Step 2: Translate Sentence into Equation

Turning the given sentence into an equation, 'a number' can be written as \(x\), 'increased by 2' can be expressed as \(+2\), and 'is 4' can be written as \(=4\). So the sentence 'A number increased by 2 is 4' should be translated into the equation \(x+2=4\).

Step 3: Match the Equation with Options

Now the goal is to compare the equation \(x+2=4\) with the options available. Looking at the options, option B represents 'A number increased by 2 is 4'. So, the correct choice is B.

Key Concepts

Algebraic Expressions

Algebraic expressions are the backbone of algebra and allow us to represent real world situations mathematically. Think of them as phrases that involve numbers, variables (like x or y), and operation symbols such as + (plus), - (minus), * (multiply), and / (divide).

For instance, in the exercise given, 'a number increased by 2' can be thought of as a phrase where 'a number' stands for a variable (let's use x) and 'increased by' suggests addition. So this phrase translates to the algebraic expression x + 2. It's crucial to understand each part of the phrase when translating to an expression because each word indicates a specific mathematical operation.

Basic Algebra

Basic algebra involves understanding and using fundamental concepts such as variables, constants, and the rules of arithmetic operations to solve equations. A variable is a symbol that represents an unknown value, while constants are fixed values that do not change. In solving algebraic problems, it is essential to treat equations like a balance scale, where your goal is to isolate the variable.

When we refer to 'a number' in the problem, we assign a variable, such as x, because its value is not yet known. The process includes performing operations 'to both sides' of the equation to maintain balance. That's why when we translate 'a number increased by 2 is 4' into x + 2 = 4, we're setting up an equation that can later be solved for x by 'balancing' both sides with inverse operations.

Equations

Equations are statements of equality that show two expressions are equal. They usually contain an equals sign (=) and are solved to find the value of the unknown variables. In the context of our exercise, an equation is a formal way to express the problem presented in words as a mathematical statement.

For the sentence 'A number increased by 2 is 4', the equation is x + 2 = 4. The left side of the equation x + 2 represents the algebraic expression we discussed earlier, and the right side, 4, represents the outcome or total value after the increase. The equation tells us that when 2 is added to some unknown number x, the result is equal to 4.

Problem-solving

Problem-solving in algebra involves a series of steps to find the value of an unknown variable. It starts with understanding the problem, as we did in step 1, by breaking down the sentence into its core elements. Next is translating these words into an algebraic language, as done in step 2.

After translating the problem, we compare our equation with provided options (step 3), seeking the one that accurately represents our translation. The key is methodically approaching each segment of the problem, using algebraic rules and logical reasoning. Once the correct equation is identified, which in our case is option B, we can proceed to solve for the unknown x by using inverse operations. Solving algebra problems this way ensures a structured approach that makes it easier to find solutions and understand the reasoning behind them.