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The substitution method is a fundamental technique in solving linear equations, allowing us to find out if a particular value is a solution to the equation. Begin by identifying the variable in the equation. In our example, the variable is represented as 's'. Once the variable is pinpointed, you can replace it with the given number to verify if it satisfies the equation. This 'plugging in' process transforms the equation from a generic form where the variable could be any number, to a specific instance that can be easily evaluated using basic arithmetic.

For instance, when we have the equation \(2s - 1 = s + 7\), and we want to determine if 6 is a solution, we substitute 's' with 6. By replacing the variable with a concrete number, we can carry out concrete arithmetic operations to check if the equation holds true. More importantly, this method can be applied to any linear equation, making it a very versatile and valuable tool in algebra.

After substituting the value into the equation, the next step is to perform arithmetic operations to simplify both sides of the equation. The goal is to reach a basic numerical statement that can be easily assessed as true or false.

In our example, with \(2(6) - 1 = 6 + 7\), we multiply 2 by 6 and then subtract 1 to simplify the left side of the equation. On the right side, we simply add 6 and 7 together. These operations - multiplication, addition, and subtraction - are the building blocks of solving equations. Mastering these skills is crucial because complex problems often involve simplifying expressions through these operations. Providing practice in breaking down equations into simpler, solvable pieces can improve understanding and build confidence in handling more challenging problems.

Verifying solutions is the final step in the process of solving linear equations. It’s essential to ensure that the value you substituted creates a true statement once the equation is simplified. After performing your arithmetic operations, compare both sides of the equation. If the left and right sides are not equal, then the value you substituted is not a solution to the equation.

In our example, once we’ve simplified the equation, we end up with \(11 ≠ 13\), which clearly shows an inequality. Therefore, we can definitively conclude that 6 is not a solution to the given equation. This act of verification is much like a detective confirming an alibi – it either holds up to scrutiny or it doesn’t. When the alibi falls apart, you know you need to look elsewhere for your solution. Similarly, verifying helps students to check their work and ensure accuracy, an invaluable skill in mathematics.