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When solving linear equations, the first goal is often to isolate the variable. This means you want to get the variable by itself on one side of the equation. In our example, the original equation is \(5 = z + 19\). To isolate the variable \(z\), we subtract 19 from both sides of the equation: \[ 5 - 19 = z + 19 - 19 \]. This step is crucial because it allows us to simplify the equation and uncover the value of \(z\). By performing the same operation on both sides, we maintain the balance of the equation. After simplifying, we have: \[ -14 = z \]. Now, \(z\) is isolated, which means we've found the solution, \(z = -14\). Remember, the isolation of variables is the foundational technique in algebra for solving equations.

Once you have isolated the variable and found a solution, it's essential to verify it. Verify means to check if the solution you’ve found actually satisfies the original equation. In our example, we found \(z = -14\). To verify: 1. Substitute \(z = -14\) back into the original equation: $$ 5 = (-14) + 19 $$ 2. Simplify the right side of the equation: $$ 5 = 5 $$ Both sides of the equation are equal, so our solution \(z = -14\) is correct. Verifying not only ensures that no mistakes were made but also builds confidence in your problem-solving skills. Always take this final step to confirm your answers.

Linear equations are a type of equation where the highest power of the variable is one. These equations form a straight line when plotted on a graph. In our example, \(5 = z + 19\) is a linear equation. Here’s what makes an equation linear:

• The variable (in this case, \(z\)) is not raised to any power other than one.
• Variables are not multiplied or divided by each other.
• The equation represents a balance between two expressions.

Recognizing and understanding linear equations is crucial because it helps you use the correct methods for solving them. These methods include isolating the variable and verifying the solution. Knowing what makes an equation linear will guide you in approaching and solving even more complex equations in the future.