Question

Solve the equation. $$9=x-4$$

Short answer

The solution to the equation \(9 = x - 4\) is \(x = 13\).

Step-by-step solution

Identify the Equation

The given equation is \(9 = x - 4\). The goal is to isolate \(x\).

Use the Addition Property of Equality

To isolate \(x\), add 4 to both sides of the equation. This gives: \(9 + 4 = x - 4 + 4\). This simplifies to: \(13 = x\).

Check the Solution

To verify if the solution is correct, substitute \(13\) for \(x\) in the original equation: \(9 = 13 - 4\). The left side (9) equals the right side (9), so the solution is correct.

Key concepts

Addition Property of Equality

The addition property of equality is a fundamental rule of algebra which states that you can add the same number to both sides of an equation without changing the equation's solution. In practice, when you come across a linear equation like \(9 = x - 4\), you can use this property to isolate the variable.

For example, to balance the equation and solve for \(x\), you would add 4 to both sides of the equation, as shown in the step by step solution. This process is like balancing scales; whatever you do to one side, you must do to the other to keep the equation in equilibrium. It's important to remember that this property applies to subtraction as well since subtraction is just adding a negative number.

Isolating Variables

Isolating the variable, often the goal in solving an equation, means rearranging the equation so the variable you're solving for is alone on one side of the equation. When dealing with linear equations, this typically involves using inverse operations to undo whatever is being done to the variable.

In the exercise given, \(x\) is being subtracted by 4. So, to isolate \(x\), you do the opposite operation—add 4 to both sides. As a result, the subtraction of 4 and the addition of 4 cancel each other on the side with the variable, leaving \(x\) by itself on one side. Always aim to isolate the variable with minimal steps to reduce the risk of errors.

Equation Solving Steps

The process of solving linear equations involves a series of steps that lead you from the problem statement to the solution. In the context of the exercise presented, these steps are clear and sequential.

Step 1 is to identify the equation you're working with and set your goal, usually isolating a particular variable. In Step 2, apply the addition property of equality to both sides to move numbers away from the variable. Once the variable is alone, Step 3 is to check your solution by substituting it back into the original equation and verifying that both sides are equal.

Pro Tip: Always Check Your Work

Never skip the checking stage. This step ensures your solution is correct and teaches you to validate your results, a hugely valuable habit in mathematics and beyond.