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Solving linear equations is a fundamental skill in algebra. A linear equation is an equation that can be written in the form of \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. The goal when solving such equations is to find the value of \( x \) that makes the equation true. There are several systematic steps and techniques used to isolate the variable and solve the equation. In general, the process involves rearranging the equation to get the variable by itself on one side of the equation, while keeping the equation balanced. This often means performing the same operation on both sides of the equation, such as adding, subtracting, multiplying, or dividing by a number. Let's go step by step to understand this further.

To solve a linear equation, we need to isolate the variable. Here’s how we can do that:

• Start with the original equation: We begin with the given equation. For example, \( -13x + 7 = -19 \).
• Perform operations to both sides: Our goal is to get the term with the variable (\( x \)) by itself. We first remove any constants added or subtracted to the variable term. Here, we subtract 7 from both sides: \[ -13x + 7 - 7 = -19 - 7 \] simplifying it to: \[-13x = -26 \].
• Isolate the variable: Next, we want \( x \) to be alone. Since -13 is multiplied by \( x \), we do the opposite (division) to isolate \( x \). We divide both sides by -13: \[ \frac{-13x}{-13} = \frac{-26}{-13} \] which simplifies to \ ( x = 2 ) \.

Following these steps helps us systematically solve linear equations.

Checking solutions is a crucial step to ensure the answer we found is correct.

Let's take the value we obtained for \( x \) and substitute it back into the original equation: \( -13(2) + 7 = -19 \). Working through this substitution:
\[-26 + 7 = -19 \], which simplifies correctly to -19. This shows our solution \( x = 2 \) is indeed correct.

Always make it a habit to verify your solution by substitution. It confirms you didn’t make any mistake in the process.
To sum up:

• Always substitute your solution back into the original equation.
• Ensure both sides of the equation equal after substitution.

This practice helps ensure the accuracy of your solutions.