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In algebra, a key step is often to simplify expressions by combining like terms. Like terms are terms that have the same variable raised to the same power. In the equation provided, we have two terms on the left side, specifically, \( 7x \) and \( 2x \). Both of these terms contain the variable \( x \) with the same exponent, making them 'like terms.'
To combine like terms, you simply add their coefficients together. For the original equation of \( 7x + 2x = 8x - 3 \), you add the coefficients 7 and 2, resulting in \( 9x \).
This simplifies our equation to \( 9x = 8x - 3 \).
Combining like terms helps make equations simpler and easier to solve. It reduces the number of terms, making it your first step whenever you spot them in an equation.

Once like terms are combined, the next important step is isolating the variable. The goal is to have the variable on one side of the equation and constants on the other.
In our equation, \( 9x = 8x - 3 \), we aim to get \( x \) by itself on one side. This is done by removing other x terms from the same side as \( x \) using inverse operations. Here, we subtract \( 8x \) from both sides to move it away from the right side:
\( 9x - 8x = 8x - 8x - 3 \). This simplifies to \( x = -3 \).
Now, \( x \) is isolated. Isolating the variable in any equation is crucial as it helps find the solution directly. It focuses our task on solving for the unknown, streamlining the process greatly.

After finding a solution to a linear equation, it's wise to verify that solution. This ensures the found value satisfies the original equation.
We found \( x = -3 \). To verify, substitute it back into the original equation \( 7x + 2x = 8x - 3 \). Substitute \( -3 \) for \( x \), which gives \( 7(-3) + 2(-3) = 8(-3) - 3 \).
Simplify both sides: Left side becomes \( -21 - 6 = -27 \) and the right side \( -24 - 3 = -27 \). Since both sides of the equation equal \( -27 \), our solution is correct.
Verification is a useful check, offering confidence that no errors were made during the earlier steps. It's a habit that can encourage accuracy in solving equations across various contexts.