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Solving linear equations is an essential skill in algebra. A linear equation is an equation in which the highest power of the variable is one. This means it forms a straight line when graphed. These equations have a straightforward structure, often written in the form of \( ax + b = c \). Here, \( x \) is the variable you solve for.
To solve a linear equation, follow these general steps:

• Isolate the variable on one side of the equation. This often involves moving constants or other terms to the opposite side.
• Perform inverse operations to simplify the equation. For example, if the equation has additions, use subtraction to eliminate them, and vice versa.
• Check your work by substituting the variable back into the equation.

For the equation \( x + 4 = 2x \), the process involves making sure both sides equal when solving for \( x \). If the equation doesn't balance with numbers plugged in, then that value isn’t a solution.

Once you have solved a linear equation, it's crucial to verify your solution to ensure it satisfies the original equation. Verification ensures accuracy because even minor mistakes can lead to incorrect results in algebra.
Verification involves substituting the value found for the variable back into the original equation. If both sides of the equation are equal with this substitution, then the solution is verified. If they are not equal, the solution is incorrect.
For example, in the exercise given:

• Substitute \( x = -1 \): The left side is \(-1 + 4 = 3\), and the right side is \(2(-1) = -2\). The sides are not equal, so \( x = -1 \) is not a solution.
• Substitute \( x = 2 \): The left side is \(2 + 4 = 6\), and the right side is \(2(2) = 4\). The sides are not equal, confirming \( x = 2 \) is not a solution.

Verification helps ensure that wherever possible, mistakes are caught and solutions are accurate.

Mathematical substitution is fundamental when evaluating whether a particular value satisfies an equation. It involves replacing the variable in the equation with a given number and simplifying the expression to see if it results in a true statement.
To perform substitution:

• Replace the variable with the chosen number.
• Calculate both sides of the equation separately.
• Compare the results: if both sides are equal, the substitution confirms a valid solution.

In our exercise, substitution of each potential solution (i.e., \( x = -1 \) and \( x = 2 \)) is done to check whether they satisfy the equation \( x + 4 = 2x \). Neither substitution resulted in both sides of the equation being equal, which means neither \( x = -1 \) nor \( x = 2 \) is a valid solution for the equation.
This technique is invaluable for solution verification and allows for a confident understanding of algebraic solutions.