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Whenever you are faced with solving a linear equation, the primary goal is to isolate the variable. This means getting the variable by itself on one side of the equation. This process allows us to determine the value of the variable that makes the equation true. In the equation \(-7 = x + 11\), the variable we need to isolate is \(x\). To achieve this, we perform the same operation on both sides of the equation. Subtracting 11 from both sides of the equation helps to undo the '+' operation paired with \(x\). By doing these steps:

• Subtract 11 from the left side: \(-7 - 11 = -18\)
• Subtract 11 from the right side: \(x + 11 - 11 = x\)

Now the equation is \(-18 = x\), which clearly shows the value of \(x\). It's like peeling away layers to find what is underneath. Always remember that what you do to one side of an equation, you must do to the other, to maintain balance.

After finding a possible solution for the variable, it's crucial to verify that this solution is correct. Verification ensures that the steps we took were right and that we didn't make any calculation errors. With the solution \(x = -18\) from our initial equation, substitute it back into the original equation and check:\(-7 = -18 + 11\)Calculating the right side gives us:

• \(-18 + 11 = -7\)

When both sides of the equation match, we know our solution is correct. It acts as a final seal of approval on your work. This step helps to catch mistakes and reassures us that \(x = -18\) truly satisfies the original equation. It’s like double-checking a math problem to ensure accuracy.

At the core of solving equations is algebraic manipulation, which involves performing operations to transform an equation while keeping it balanced. This is key to isolating the variable. Techniques include addition, subtraction, multiplication, or division.In our example, the manipulation starts with subtracting 11 from both sides of \(-7 = x + 11\). Here’s how you manipulate equations:

• Identify the operations associated with the variable (e.g., +11 with \(x\)).
• Apply the inverse operation to remove these associations, like subtraction for addition.
• Keep the equation balanced by doing the same operation to both sides.

This manipulation helps reshape the equation into a simpler form that reveals the solution, \(x = -18\). Algebraic manipulation is a basic yet powerful tool in mathematics, giving us control to solve and understand equations.