Problem 12 - Algebra A Combined Function

Question

Solve. The sum of two numbers is 76 and their difference is 52. Find the two numbers.

Step-by-Step Solution

Answer

The two numbers are 64 and 12.

Step 1: Set Up the Equations

Let's denote the two numbers as \(x\) and \(y\). According to the problem, the sum of the two numbers is 76, so we have the equation \(x + y = 76\). Next, the difference of the two numbers is 52, which gives us the equation \(x - y = 52\).

Step 2: Solve for One Variable

We can solve these equations simultaneously. Start by adding the two equations. \[(x + y) + (x - y) = 76 + 52\] This simplifies to \[2x = 128\]. Now, solve for \(x\) by dividing both sides by 2, \[x = 64\].

Step 3: Solve for the Second Variable

Now that we have \(x = 64\), substitute \(x\) into any of the original equations to solve for \(y\). Using \(x + y = 76\), we have \[64 + y = 76\]. Solve for \(y\) by subtracting 64 from both sides, \[y = 12\].

Step 4: Verification

To ensure the solution is correct, check that both original conditions are met. The sum \(x + y = 64 + 12 = 76\) and the difference \(x - y = 64 - 12 = 52\). Both conditions are satisfied.

Key Concepts

Algebraic Equation

Understanding algebraic equations is essential when solving problems that involve unknown numbers. In an algebraic equation, we use variables like \(x\) and \(y\) to represent unknowns.
A key step in solving a problem involving two unknowns, such as in this exercise, is setting up the right equations. Here, we are given two key clues:

• The sum of two numbers is 76.
• Their difference is 52.

Using these clues, we form two equations:

• \(x + y = 76\)
• \(x - y = 52\)

These equations connect the variables together in a way that allows us to systematically find their values. The art of forming such equations is a fundamental skill it allows us to translate vague word problems into precise mathematical language.

Problem Solving

Solving simultaneous equations might seem tricky, but it's a step-by-step process. It involves finding the values of the variables that satisfy both equations. We start by manipulating the equations: For the equations \(x + y = 76\) and \(x - y = 52\), notice how adding them cancels out \(y\). Doing so simplifies the system and helps isolate one variable:

• Add the equations: \((x + y) + (x - y) = 76 + 52\).
• This simplifies to \(2x = 128\).
• Solve for \(x\) by dividing both sides by 2, resulting in \(x = 64\).

By focusing on one variable first, we reduce the complexity and find one part of the solution. This method of solving equations, often termed 'elimination technique,' is a powerful tool in algebra that breaks problems into manageable parts.

Mathematical Verification

Once we have potential solutions for our variables, it's important to verify that these solutions literally check out. Verification ensures no mistakes were made in calculations.
Substitute the values back into the original problem conditions to check your work:

• We found \(x = 64\) and substituted \(x\) back into \(x + y = 76\) to solve for \(y = 12\).
• Double check by ensuring both conditions: \(x + y = 64 + 12 = 76\) and \(x - y = 64 - 12 = 52\) are satisfied.

Both equations hold true, reaffirming our solution's accuracy. Verification is the step where confidence is built. It's not just about getting the answer but confirming that it's the correct answer—a crucial habit in problem-solving.