Problem 63 - Elementary Algebra Within Reach

Question

Checking a Solution Explain how to check a solution of an equation. Then illustrate your explanation using the equation \(3 x+2=11\).

Step-by-Step Solution

Answer

The solution of the equation \(3x + 2 = 11\) is \(x = 3\), and checking it by substituting \(x = 3\) back into the original equation shows that it is a valid solution.

Step 1: Find the value of the variable

The primary task is to solve the given equation \(3x + 2 = 11\). For this, first subtract 2 from both sides to get \(3x = 9\). Then, divide both sides of the equation by 3 to get the value of the variable \(x\). This gives \(x = 3\).

Step 2: Substitute the value back into the equation

To check if the obtained value is accurate, substitute \(x = 3\) back into the original equation \(3x + 2 = 11\). This yields \(3*3 + 2\).

Step 3: Perform the operations

The expression \(3*3 + 2\) simplifies to 9 + 2, which equals 11. Since the left side of the equation equals the right side when \(x = 3\), this confirms that the solution is correct.

Key Concepts

Solving Linear Equations

Solving linear equations involves finding the value of the variable that makes the equation true. In a linear equation like \(3x + 2 = 11\), our goal is to isolate the variable, typically \(x\), on one side of the equation. Here's how you can do it:

• First, identify the operations affecting the variable. In our equation, \(3x\) is affected by the addition of 2.
• We perform the inverse of these operations to move them to the other side of the equation. Here, we subtract 2 from both sides to counteract the addition. This leaves us with \(3x = 9\).
• Now, \(x\) is multiplied by 3. So, divide both sides by 3 to get \(x = 3\).

Breaking the equation down into simpler steps makes finding the solution much more manageable. By performing one-step operations repeatedly, we gradually isolate the variable.

Checking Solutions

After obtaining a solution, it's essential to verify its correctness. This process is called checking solutions. It's like proof-reading your own work to make sure everything adds up.

• To check, substitute the found value of the variable back into the original equation.
• Ensure both sides of the equation remain equal.

Using our solved equation \(3x + 2 = 11\) with \(x = 3\), we substitute \(x\) with 3, giving us \(3 \cdot 3 + 2\). This evaluates to 9 + 2 which equals 11 – the original right side of the equation. Since both sides match, our solution is correct.

Substitution Method

The substitution method is a critical step for checking solutions. Here's how it works:

• Take the value you calculated (in this case, \(x = 3\)).
• Replace the variable in the original equation with this value.
• Solve the equation to see if both sides are equal.

For \(3x + 2 = 11\), substituting \(x\) with 3 gives us \(3 \cdot 3 + 2\). Simplifying this, we find that it equals 11, which is what we started with on the right side of the equation. Substitution allows us to confirm that our initial solution is indeed correct.

Basic Arithmetic Operations

Basic arithmetic operations are fundamental in solving algebra equations. They include addition, subtraction, multiplication, and division, shaping how we manipulate and solve equations.

• Addition and subtraction are often used to move constant terms from one side of an equation to the other.
• Multiplication and division are used primarily to eliminate coefficients that are attached to variables.

In solving \(3x + 2 = 11\), we used subtraction to remove 2 from the left side, resulting in \(3x = 9\). Then we used division to isolate \(x\), which gave us \(x = 3\). Mastery of these operations is essential and paves the way to more complex mathematical problem-solving!