Problem 22 - Elementary Algebra

Question

For the following problems, translate the following phrases or sentences into mathematical expressions or equations. Twice a number is eleven.

Step-by-Step Solution

Answer

Question: Translate the following phrase into a mathematical equation: "Twice a number is eleven." Answer: 2x = 11

Step 1: Identify the unknown variable.

Let x represent the unknown number we want to find.

Step 2: Express the given phrase mathematically.

The phrase states that "twice a number is eleven", which means we need to multiply the unknown number (x) by 2 and set it equal to 11. This can be written as an equation: 2x = 11. Now we have translated the phrase into a mathematical equation: 2x = 11.

Key Concepts

Algebraic Expressions

Understanding algebraic expressions is vital in solving many types of mathematical problems. An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. Variables are symbols, usually letters, that stand in for unknown values and can change or vary. For example, the expression twice a number can be represented as 2x, where x is the variable representing the number.

To translate phrases into algebraic expressions, follow these steps:

• Identify any numerical operations such as 'twice' (which means 'multiplied by 2').
• Recognize the variable that the operation is referring to.
• Combine the operation and the variable to create an expression.

Remember, expressions do not have an equals sign, while equations do. It's a common mistake to confuse the two!

Mathematical Equations

A mathematical equation is a statement that asserts the equality of two expressions. Equations are essential tools for solving problems and finding unknown values. They are usually written with an equals sign (\(=\)). For instance, in our exercise, 'Twice a number is eleven' leads to the equation 2x = 11.

To turn phrases into equations:

• Identify the algebraic expression from the phrase.
• Determine the 'is' part of the phrase, which usually indicates equality.
• Set the algebraic expression equal to the value stated in the phrase.

Equations can often be simplified or manipulated using various algebraic principles to solve for the variable, in this case, the unknown number that 'x' represents.

Problem Solving in Algebra

Effective problem solving in algebra is a process that involves understanding the problem, translating it into mathematical language, and then methodically simplifying and solving the equations or expressions.

To improve your problem-solving skills in algebra, consider these tips:

• Take the time to fully understand the problem before trying to solve it.
• Break down the problem into smaller, more manageable parts if possible.
• Translate words into algebraic expressions or equations as we've done with 'Twice a number is eleven', which becomes 2x = 11.
• Use algebraic rules to simplify and manipulate the equations to solve for the variable.
• Check your answer by plugging it back into the original equation to ensure it satisfies the condition.

Persistence and practice are key components to becoming proficient at solving algebra problems.