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When solving linear equations such as \( 8x = 7x - 3 \), the goal is to find the value of \( x \) that makes the equation true. This involves isolating the variable \( x \) on one side of the equation. To do this, identify which terms have the variable you need to solve for.
In our example, we notice both sides have an \( x \) term. To get all variable terms on one side, perform the operation that moves terms without changing the equation's balance. Here, subtracting \( 7x \) from both sides we can simplify this process.
Once terms not involving \( x \) are on the other side, simplify to find \( x = -3 \). Successfully isolating the variable is a crucial first step in solving any equation.

Step-by-step solutions allow us to methodically approach and solve equations. Each step builds on the last, ensuring clarity and ease of understanding. Let's break it down:

• **Begin by identifying the variable:** Look for terms with \( x \) on both sides of the equation.
• **Use inverse operations:** To bring similar terms together, use operations that cancel out terms. Here, subtract \( 7x \) from both sides to gather the \( x \) terms on one side.
• **Simplify the equation:** Once moved, simplify the terms to eliminate extra numbers, making the equation as straightforward as possible.

With practice, each step becomes second nature, allowing for efficient and effective problem solving.

Verifying solutions is an essential final step in solving equations. It confirms that your solution is correct and consistent with the original equation. Here's how it's done:

• **Substitute back:** Plug the value of \( x \) found from your solution back into the original equation. Example: for \( x = -3 \), substitute into \( 8x = 7x - 3 \), giving \[ 8(-3) = 7(-3) - 3 \].
• **Simplify both sides:** Perform all the arithmetic to simplify both sides of the equation and ensure they are equal.
• **Confirm equality:** If both sides equal, your solution is verified as correct. Here, both sides became \( -24 \), proving \( x = -3 \) is correct.

Verification is like a safety check, ensuring no errors slipped through the solution process, and it guarantees your answer is accurate.