Problem 37 - Elementary Algebra

Question

For the following problems, translate the following phrases or sentences into mathematical expressions or equations. A number plus seven is divided by two and the result is twenty-two.

Step-by-Step Solution

Answer

Question: Translate the following sentence into a mathematical equation and find the unknown number: "A number plus seven is divided by two, and the result is twenty-two." Answer: The unknown number is 37.

Step 1: Identify the unknown variable and the operations

Let 'x' be the unknown number. According to the sentence, we will perform two operations: adding seven to the number and then dividing the result by two.

Step 2: Translate the sentence into a mathematical equation

Given that a number (x) plus seven is divided by two, and the result is twenty-two, we have: \[\frac{x+7}{2} = 22\]

Step 3: Solve the equation to find the unknown number

Now, solve for 'x': \[\frac{x+7}{2} = 22 \Rightarrow x+7 = 2 \times 22\] \[x+7 = 44 \Rightarrow x = 44-7\] \[x = 37\] The unknown number is 37.

Key Concepts

Algebraic Equations

When we talk about algebraic equations, we're referring to mathematical statements that show the balance between two expressions separated by an equal sign (\(=\)). An equation tells us that the value of the expression on one side is the same as the value on the other side, no matter what numbers the variables might represent.

For instance, the equation \(\frac{x+7}{2} = 22\) from our exercise is an algebraic equation. Equations can be simple, with only one variable like 'x', or they can be more complex with multiple variables. But regardless of complexity, the goal remains the same: to find the value(s) of the unknown(s) that make the equation true.

In solving an equation, we're essentially playing a game of balance. We perform operations on both sides of the equation to isolate the variable, taking care to do the same thing to both sides to maintain the equation's balance. This approach leads us toward the solution, as shown in the step-by-step solution provided. This kind of problem-solving strategy is fundamental in algebra and has applications in many fields including science, engineering, and economics.

Mathematical Translation

Mathematical translation is the process of converting words, phrases, or real-world scenarios into an algebraic language of numbers and symbols that can be manipulated and solved. This often involves recognizing key terms that correspond to mathematical operations. For example, 'plus' translates to addition (\(+\)), 'divided by' to division (\(\div\) or \(\frac{\cdot}{\cdot}\)), and 'result is' to an equals sign (\(=\)).

This skill is vital because it enables us to take a verbal description and create an algebraic representation that can be mathematically analyzed. If we refer back to the exercise, the phrase 'a number plus seven is divided by two and the result is twenty-two' translates to \(\frac{x+7}{2} = 22\) when we know that 'a number' is our variable 'x'.

Improving in this area involves practice and familiarity with algebraic vocabulary. Sometimes, drawing diagrams or using tables can help visualize the problem before translating it into equations. Ensuring that the translation captures all elements of the original statement correctly is crucial to finding the right solution.

Solving Algebraic Equations

Solving algebraic equations is about finding the value or values that satisfy the equation - making it a true statement. The solving process usually involves simplifying the equation and isolating the variable through various algebraic operations, such as addition, subtraction, multiplication, and division.

In the example \(\frac{x+7}{2} = 22\), we start by clearing the fraction by multiplying both sides by 2, which is the denominator. Then, we simplify the equation to \(x + 7 = 44\) by performing the same operation on both sides of the equals sign. This process of maintaining the balance is essential as it ensures the equation remains valid.

Next, we isolate 'x' by subtracting 7 from both sides, leading us to the answer \(x = 37\). Throughout each step, we carried out precise operations to maintain equality, which is a core principle in solving algebraic equations.

To get better at solving these equations, it's important to practice regularly and to learn from mistakes by reviewing missteps and correcting them. Understanding the 'why' behind each operation rather than just memorizing steps will enable students to tackle a wide variety of algebraic challenges.