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Linear equations are like simple puzzles that involve finding the value of the variable that makes the equation true. In a linear equation, the variable is always raised to the power of one. For example, in the equation \(y + 3 + 2y - 4 = 6\), the variable is \(y\), and there are no exponents or complicated operations.

To solve these equations, you need to simplify and rearrange terms. The aim is to find what value of the variable makes both sides of the equation equal. Follow these general steps to solve linear equations:

• Simplify each side by combining like terms.
• Isolate the variable term.
• Solve for the variable by performing basic arithmetic operations.

Each of these steps helps get closer to finding the value of the unknown variable, which balances the equation.

When solving linear equations, one of the most important steps is to combine like terms. "Like terms" are terms in the equation that contain the same variable raised to the same power. For example, in \(y + 2y\), both terms have the variable 'y'. So you can combine them.

Combining like terms simplifies the equation and helps pave the way towards finding the solution. Here's how it works in the equation \(y + 3 + 2y - 4 = 6\):

• Add \(y\) and \(2y\): \(y + 2y = 3y\).
• Then group the constant numbers: \(3 - 4 = -1\).

This process turns the equation into \(3y - 1 = 6\), which is simpler and easier to solve. Remember, combining like terms is all about making the equation less cluttered and more manageable.

Isolating the variable is an essential step in solving linear equations, allowing you to identify the value that makes the equation true. After simplifying the equation by combining like terms, your goal is to get the variable by itself on one side of the equation.

In the equation \(3y - 1 = 6\), you isolate the term with 'y' by adding 1 to both sides, to cancel out the \(-1\):

\[3y - 1 + 1 = 6 + 1\]
\[3y = 7\]

Once you have \(3y = 7\), you divide both sides by 3 to solve for \(y\):

\[y = \frac{7}{3}\]

Isolating the variable is crucial because it gets you closer to solving the equation. You are effectively undoing any operations that are affecting the variable, to reveal what value of the variable will satisfy the equation.