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Solving linear equations often begins with isolating the variable. This step is crucial because it simplifies the equation, allowing you to find the value of the unknown variable. Isolation means to get the variable by itself on one side of the equation with everything else on the other.

Consider the example equation from the exercise,
\[ 4y + 3 = 11 \].

The goal is to isolate y. Start by eliminating all other terms that aren’t attached to y from its side of the equation. That’s done by subtracting 3 from both sides, which results in
\[ 4y = 8 \].

Now, the variable y is the only term on the left side of the equation, successfully isolated. The equation has become simpler, setting the stage for finding the variable's value.

Once the variable is isolated, the next step is simplifying the equation. This typically involves performing basic arithmetic operations to modify the equation into a form where the variable has a coefficient of 1. Simplification makes it easier to identify the solution.

In the exercise,
\[ 4y = 8 \],

the variable y is multiplied by the coefficient 4. To simplify, divide both sides of the equation by 4 - the same number as the coefficient. This yields
\[ y = 2 \].

The equation is now simplified to its most basic form. It’s important to always perform the same operation on both sides to maintain the balance of the equation. When simplifying, use inverse operations—addition with subtraction and multiplication with division—to eliminate or modify coefficients and constants effectively.

Checking the solution is an essential part of solving equations. It involves substituting the obtained variable value back into the original equation to ensure its correctness. A valid solution will satisfy the original equation, with both sides equal to each other.

From the exercise, after finding \( y = 2 \),

you must check it by replacing y in the initial equation
\( 4y + 3 = 11 \).

This process results in
\[ 4(2) + 3 = 11 \],
and simplifies to
\[ 11 = 11 \].

Since both sides are equal, the solution y = 2 is confirmed to be correct. Always remember to check your solutions. This will not only verify your answer but also reinforce a greater understanding of how to manipulate and solve linear equations.