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The substitution method is a way to solve systems of equations by replacing one variable with an expression involving the other variable. This technique works effectively for linear equations. Here is a step-by-step explanation:

First, solve one of the equations for one of the variables. In this example, we solve the second equation, \(2x + 5y = -3\), for \(x\):

\[2x = -3 - 5y \] \[ x = \frac{-3 - 5y}{2} \]
Next, substitute this expression for \(x\) in the other equation, allowing you to solve for one unknown.

This method reduces complications when dealing with systems, making it possible to simplify and solve linear equations accurately.

Solving linear equations involves isolating the variable to find its value. Here are the steps you need to follow:

After substitution, we had:
\[ 3\frac{-3 - 5y}{2} + 8y = -3 \]
To solve for the variable, simplify the equation step-by-step. Perform operations like addition, subtraction, multiplication, and division to express the variable explicitly.

Continuing from our example:
\[ \frac{3(-3 - 5y)}{2} + 8y = -3 \]
Multiply both sides by 2 to eliminate the fraction:
\[ -9 - 15y + 16y = -6 \]
Combine like terms and solve for \(y\).
Finally, substitute \(y\) back into one of the original equations to solve for \(x\). This step-by-step approach ensures a logical simplification process.

Combining like terms simplifies equations by merging similar variable terms. This is crucial in linear equations to isolate the variable:

In our exercise, after substitution, we had:
\[ \frac{-9 - 15y}{2} + 8y = -3 \]
We multiplied all terms by 2 to remove the fraction:
\[ -9 - 15y + 16y = -6 \]
Here, \( -15y \) and \( 16y \) are like terms. Combine these terms:
\[ -15y + 16y = y \]
This simplifies the equation to:
\[ -9 + y = -6 \]
Combining like terms simplifies the problem, making it easier to isolate and solve for the variable.

Eliminating fractions helps streamline equations, making them easier to solve. Multiply every term by the denominator to achieve this:

Starting with:
\[ 3\frac{-3 - 5y}{2} + 8y = -3 \]
To eliminate the fraction, multiply by 2:
\[ -9 - 15y + 16y = -6 \]
The fractions are now gone, simplifying the equation and reducing potential errors:
\[ -9 + y = -6 \]
This process turns complex fractional equations into simpler algebraic forms, supporting accuracy in solving systems of linear equations. Properly managing fractions at this stage ensures you maintain valid steps across your work.