Problem 76 - Introductory Algebra for College Students

Question

Write each sentence as an equation. Let the variable \(x\) represent the number. Evaluate \(\frac{x-y}{3}\) when \(x\) is 2 more than 5 times \(y\) and \(y=4 .\)

Step-by-Step Solution

Answer

So the value of \(\frac{x-y}{3}\) when \(x = 5(4) + 2=22\), \(y=4\) equals \(6\).

Step 1: Formulate the first equation

Translate '2 more than 5 times \(y\)' into an algebraic expression to represent \(x\). This can be written as \(x = 5y + 2\). So this is the equation from the first sentence.

Step 2: Substitute \(y\) into the first equation

Now plug in the given value of \(y\), which is \(4\), into the equation \(x = 5y + 2\). This gives \(x = 5(4) + 2 = 20 + 2 = 22\). Now we know the values of both \(x\) and \(y\).

Step 3: Evaluate the expression \(\frac{x-y}{3}\)

Substitute the found values for \(x\) and \(y\) into \(\frac{x-y}{3}\). This leads to: \(\frac{22-4}{3} = \frac{18}{3} = 6\)

Key Concepts

Understanding Equations

Equations are like mathematical sentences that show the relationship between different quantities. They use an equal sign to indicate that two expressions are equal.
In this exercise, we were given a verbal sentence: "2 more than 5 times \(y\)". Our task was to translate this into an algebraic equation. The equation becomes \(x = 5y + 2\), meaning that \(x\) is the result of multiplying \(y\) by 5 and then adding 2.
Key things to remember about equations:

• They always have an equal sign \(=\) that separates the two sides.
• Each side can include numbers, operations, and variables.
• They represent a balance – whatever you do to one side, you must do to the other to keep it equal.

Practicing translating words into equations is key to solving these types of math problems.

The Role of Variables

Variables serve as placeholders in algebraic expressions and equations. They're like the ingredients in a recipe; they let us explore unknown quantities.
In this problem, \(x\) and \(y\) are variables. We initially don't know their values but use other information from the problem to find them. For this problem, we're told "\(y = 4\)", so we can substitute 4 for \(y\) in the equation.
Points to keep in mind about variables:

• They can be represented by any symbol, but commonly by letters like \(x, y, z\).
• They allow us to generalize mathematical principles.
• They can turn into specific numbers when given more information.

Variables help make equations adaptable to many situations, helping us solve not just one problem but many others that might be structured the same way.

Evaluation of Expressions

Evaluation is the process of finding the value of an expression when the values of variables are specified. We substitute the known values into the expression and simplify.
In our exercise, we needed to evaluate \(\frac{x-y}{3}\). After finding \(x=22\) and knowing \(y=4\), we substitute these values into the expression to get \(\frac{22-4}{3}\). This simplifying process involved performing the arithmetic operations to reach the final result, \(6\).
Steps to evaluate expressions include:

• Substitute the values given in the problem for their corresponding variables.
• Follow the order of operations, often remembered as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
• Simplify to find a single numerical value.

Understanding and practicing these steps strengthens problem-solving skills and helps in tackling more complex expressions over time.