Problem 12 - Elementary Algebra

Question

Set up an algebraic equation and then solve. If a larger integer is 2 more than 4 times another integer and their difference is \(32,\) find the integers.

Step-by-Step Solution

Answer

The integers are 10 and 42.

Step 1: Define the variables

Let the smaller integer be \( x \). Then, the larger integer can be represented as \( 4x + 2 \) since it is stated as 2 more than 4 times the smaller integer.

Step 2: Set up the equation

According to the problem, the difference between the larger integer and the smaller integer is \( 32 \). We can express this as the equation: \((4x + 2) - x = 32\).

Step 3: Simplify the equation

Simplify the equation from the previous step: \(4x + 2 - x = 32\). This simplifies to \(3x + 2 = 32\).

Step 4: Solve for \( x \)

Subtract 2 from both sides of the equation: \(3x = 30\). Divide both sides by 3 to find \( x \): \(x = 10\).

Step 5: Determine the larger integer

Substitute \( x = 10 \) back into the expression for the larger integer: \(4x + 2 = 4(10) + 2 = 40 + 2 = 42\).

Key Concepts

Integer Problems

Understanding integer problems can unlock a wide array of mathematical concepts. An integer problem often involves finding whole numbers that satisfy certain conditions or rules, like our exercise does.
When tackling integer problems, it is crucial first to identify what the integers represent and how they relate to each other in context.
In this exercise, the task is to identify two integers based on the relationship defined in the problem.

• The larger integer is said to be '2 more than 4 times' the smaller one.
• The difference between these integers is 32.

The clarity of these conditions lays the groundwork for setting up an equation. This involves recognizing how one integer is expressed in terms of another, which brings us to the next important concept: setting up equations.

Solving Equations

Once you have set up your equation, the next step is to solve it. Solving equations involves manipulating the algebraic expressions to find the value of the unknown variable.
In mathematical problems, especially those involving integers, solving equations typically means simplifying the given expressions to end up with a straightforward equation where the variable is isolated on one side.First, simplify any expressions given in the equation. For example, if your equation is \((4x + 2) - x = 32\), you begin by consolidating like terms to reduce the equation to a simpler form:

• Combine terms involving \(x\): \(4x - x = 3x\)
• The equation becomes \(3x + 2 = 32\)

Next, isolate the variable by performing inverse operations:

• Subtract 2 from both sides: \(3x = 30\)
• Divide by 3: \(x = 10\)

By solving this, you find the smaller integer. Double-checking your work ensures the values satisfy the problem's conditions.

Setting Up Equations

Setting up equations is a fundamental step in solving problems involving relationships between quantities. This process requires transforming given conditions into mathematical statements. In this context, it involves expressing the relationship between two integers as an algebraic equation.
Start by defining your variables based on the problem's description. For instance:

• Let the smaller integer be \(x\).
• The larger integer, being 2 more than 4 times the smaller integer, is represented as \(4x + 2\).

The problem also mentions that the difference between these integers is 32. To express this mathematically, you would set up the equation:

• \((4x + 2) - x = 32\)

Setting up the equation properly is crucial as it transforms the word problem into a solvable math problem. This enables you to apply algebraic methods to find the solution. Ensure that the relationship and conditions are accurately captured in your equation to provide a clear path to the solution.