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In solving linear equations, one of the first crucial steps is to isolate the variable. This means rearranging the equation so that the variable you are solving for is on one side of the equation and everything else is on the other side. In our example equation, \(-12x - 15 = 21\), the variable term is \(-12x\). To isolate it:
• Add 15 to both sides: \(-12x - 15 + 15 = 21 + 15\)
• The equation simplifies to \(-12x = 36\)

By moving the constant term (which is -15 in this case) away from the variable term, you make it easier to solve for the variable. This process involves basic arithmetic operations such as addition, subtraction, multiplication, or division. Each step brings you closer to having the variable by itself.

Always remember to perform the same operation on both sides of the equation to maintain the equality.

After solving for the variable, it is essential to check your solution to ensure it is correct. Substituting the solution back into the original equation confirms if it satisfies the equation. Let’s verify the solution for our example:
• After isolating and solving, we found \(x = -3\)
• Substitute \(x = -3\) back into the original equation: \(-12(-3) - 15 = 21\)
• Simplify the left-hand side: \(36 - 15 = 21\)

Since both sides of the equation are equal, the solution \(x = -3\) is correct. Checking the solution helps you catch potential errors and boosts your confidence in solving equations. It's a foolproof way to ensure that the work done is accurate.

Do not skip this step, as it is your final evaluation of the problem.

To solve linear equations successfully, it is helpful to follow a step-by-step process. Breaking down the steps will make the equation less overwhelming. Here’s a quick recap using our example:

Step 1: Isolate the variable term:
• Add 15 to both sides: \(-12x - 15 + 15 = 21 + 15\)
• This simplifies to \(-12x = 36\)

Step 2: Solve for the variable:
• Divide both sides by -12: \(-12x / -12 = 36 / -12\)
• This gives: \(x = -3\)

Step 3: Check the solution:
• Substitute \(x = -3\) back into the original equation: \(-12(-3) - 15 = 21\)
• Simplify: \( 36 - 15 = 21 \)
• Since both sides are equal, \(x = -3\) is the correct solution.

By taking it step by step, each part of the problem becomes manageable, making the overall process easier and more intuitive. Remember, practice makes perfect, so keep working through problems to build your confidence and skills.