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Isolating the variable in a linear equation is the process of getting the variable by itself on one side of the equation. This generally involves undoing any addition or subtraction first, then dealing with any multiplication or division.

In our example, \(-39 = 1 + 8x\), we subtract 1 from both sides to start isolating the variable term. This simplifies the equation step by step until the term involving the variable is alone on one side of the equation.

• Subtract 1: \(-39 - 1 = 1 + 8x - 1\)
• Simplify: \(-40 = 8x\)

Once we have the term with the variable isolated, we proceed to solve for the variable.

Checking your solution is critical. This ensures that the value you found for the variable actually satisfies the original equation. This can be done by substituting the value back into the original equation.

For our equation \(-39 = 1 + 8x\), we solved for \(x\) and found \(x = -5\). To validate this solution:

• Substitute \(x = -5\) into the original equation: \(-39 = 1 + 8(-5)\)
• Simplify both sides: \(-39 = 1 - 40\)
• Check: \(-39 = -39\)

Since both sides of the equation are equal, it confirms the solution is correct.

When solving linear equations, you may encounter situations where no solution exists or where an infinite number of solutions exist. These are called contradictions and identities, respectively.

A contradiction happens when you end up with a false statement, such as \(5 = 0\). This means there is no value for the variable that will satisfy the equation.

An identity occurs when you end up with a true statement, such as \(4 = 4\), after simplifying both sides of the equation. This implies that the equation is true for all values of the variable.

In our example, the final statement \(-39 = -39\) is a true statement specific to \(x = -5\). There are no contradictions or identities here. We have a specific solution for \(x\).