Problem 21 - Elementary Algebra

Question

Solve. $$ x+212=313 $$

Step-by-Step Solution

Answer

x = 101

Step 1: Understand the Equation

The given equation is a simple linear equation in the form of \(x + 212 = 313\). Our goal is to find the value of \(x\) that makes the equation true.

Step 2: Isolate the Variable

To solve for \(x\), we need to isolate \(x\) on one side of the equation. This can be done by subtracting 212 from both sides. This operation gives us the modified equation:\[x = 313 - 212\]

Step 3: Perform the Calculation

Now, we perform the subtraction on the right-hand side:\[x = 313 - 212 = 101\]

Step 4: Verify the Solution

To ensure our solution is correct, substitute \(x = 101\) back into the original equation:\[101 + 212 = 313\]Since both sides are equal, \(x = 101\) is confirmed to be the correct solution.

Key Concepts

Isolating the Variable

When solving linear equations, one of the primary steps is to isolate the variable. In the given equation, this variable is represented by \(x\). Isolating \(x\) means getting it alone on one side of the equation, which gives a clear understanding of its value. Here are a few key steps to achieve this:

• Identify the operations performed on the variable. In this case, 212 is added to \(x\).
• Use the opposite operation to cancel out extra numbers around the variable. Since 212 is added, we will use subtraction.
• Ensure that any operation done to one side of the equation is also done to the other side. This is crucial for maintaining the balance of the equation.

By subtracting 212 from both sides, we effectively isolate \(x\), simplifying our equation to \(x = 313 - 212\). This means \(x\) is now solvable.

Verifying the Solution

After finding a potential solution to a linear equation, it's important to verify it. Verification is like a double-check to ensure no mistakes were made along the solving process. Here's how you can verify a solution:

• Take the solution you found, in this case, \(x = 101\), and substitute it back into the original equation.
• Perform the arithmetic to see if both sides of the equation are equal. For our example, substitute \(x = 101\) into \(x + 212 = 313\), becoming \(101 + 212 = 313\).
• If both sides of the equation equal the same number, the solution is verified as correct. If not, reevaluate the steps to find where the error might have occurred.

Verification ensures accuracy, confirming that \(x = 101\) truly makes the original equation valid.

Simple Subtraction

Subtraction is one of the basic arithmetic operations, and understanding it plainly is vital in solving equations. In our case, subtracting numbers helps isolate the variable. Here's why subtraction is used in solving
equations:

• Subtraction helps us "undo" addition, effectively removing numbers that are grouped with the variable.
• Keep the operation straightforward. Simple subtraction involves lining up numbers by place value and carefully subtracting them digit by digit.
• Ensure accuracy by "borrowing" if needed, which occurs when a smaller digit is subtracted from a larger digit. This keeps the operation neat and correct.

In this exercise, subtracting 212 from 313 didn't require "borrowing," making the calculation straightforward, leading us directly to \(x = 101\), showing that understanding simple subtraction can make equations much less daunting.