Problem 22 - Elementary Algebra Within Reach

Question

Solve the equation and check your solution. $$3 x+21=0$$

Step-by-Step Solution

Answer

The solution to the equation \(3x+21=0\) is \(x=-7\).

Step 1: Isolate the Variable

In order to isolate the variable \(x\), subtract 21 from both sides of the equation. This results in the equation \(3x = -21\).

Step 2: Solve for the Variable

To solve for \(x\), divide both sides of the equation by 3, which gives \(x = -21 / 3\).

Step 3: Simplify the Result

By simplifying the division, the solution is found to be \(x = -7\).

Step 4: Check the Solution

Finally, check the solution by inserting \(x = -7\) into the original equation. You should end with \(3(-7) + 21 = 0\), which simplifies to 0 = 0. This confirms that the solution is correct.

Key Concepts

Isolating Variables

Isolating the variable is one of the first and most crucial steps when solving a linear equation. The goal here is to get the unknown variable, often represented as \(x\), on one side of the equation by itself. This makes it easier to solve for its value. In our exercise, we started with the equation \(3x + 21 = 0\). The first step in isolating \(x\) was to eliminate the constant term on the same side as \(x\).

To do this, we subtract 21 from both sides of the equation:

• Equation: \(3x + 21 - 21 = 0 - 21\)
• Simplified: \(3x = -21\)

At this point, \(x\) is still multiplied by 3. To continue isolating \(x\), we divide both sides by 3. This results in \(x = -7\). Once \(x\) is by itself, you can see that the equation has been solved for \(x\). By isolating \(x\) correctly, we ensure that we can find the correct value.

Checking Solutions

After solving a linear equation, it's always crucial to verify that your solution is correct. This process is known as checking the solution and provides confidence that you haven't made any mistakes along the way. In the given exercise, after isolating \(x\) and solving the equation, we found that \(x = -7\).

To check if this value is indeed correct, substitute it back into the original equation:

• Original Equation: \(3x + 21 = 0\)
• Substitute \(x = -7\): \(3(-7) + 21 = 0\)
• Calculate: \(-21 + 21 = 0\)

The left-hand side simplifies correctly to 0, which matches the right-hand side. Since both sides are equal, the solution \(x = -7\) has been successfully verified. This step is invaluable for ensuring accuracy in your calculations.

Simplifying Equations

Simplifying equations is an essential part of solving mathematical problems, especially linear equations. Simplification involves reducing an equation to its simplest form, making it easier to solve and understand. When working through the provided exercise, you first performed operations to manipulate the equation into a simpler state.

Once the equation has been altered down to \(3x = -21\), the next step is to handle any operations on the variable \(x\). Here, you divide by 3 to simplify the equation further and solve for \(x\):

• Equation before simplification: \(3x = -21\)
• Divide both sides by 3: \(x = -21 / 3\)
• Simplified to: \(x = -7\)

This kind of simplification ensures that the solution process is straightforward and the final answer is clear and easy to interpret. Simplifying equations helps prevent complications that can arise from more complex or cluttered equations.