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Solving linear equations involves finding the values of the variables that make both sides of the equation equal. Linear equations are mathematical statements in which the highest power of the variable is one. To solve these equations, we use various methods such as substitution, elimination, and graphical methods. In this exercise, we focus on solving the system of linear equations by the substitution method.
Our goal is to find the values for variables g and f that satisfy both equations simultaneously.
We start by manipulating the equations to isolate variables and substitute them back into the other equations.
This process involves several smaller steps to reach the final solution. Let's break it down to make it easier to understand.

The first step in solving the given system of equations using substitution is to isolate one variable in one of the equations. Isolating a variable means getting it alone on one side of the equation.
In the given system:

• Equation 1: \(g = 3f - 2\)
• Equation 2: \(f = g + 1\)

Equation 2 is already isolated for f, expressing f in terms of g. This simplifies our task as we can directly use this isolated variable in the next step. By isolating a variable, we simplify the complexity and make the equation easier to work with for substitution.

Simplification is a critical step in solving any algebraic equation. When you substitute the expression for one variable into the other equation, you often end up with a more complicated equation.
For example, after substituting \(f = g + 1\) from Equation 2 into Equation 1, we get: \(g = 3(g + 1) - 2\).

Distributing and simplifying: \(g = 3g + 3 - 2$$g = 3g + 1\).
During simplification, ensure you correctly combine like terms and perform arithmetic operations systematically.
It helps prevent errors and keeps the equation manageable. Breaking it down into smaller parts, as shown, can often make the simplification process easier to follow.

Substitution in algebra involves replacing one variable with an equivalent expression. This method is highly useful when dealing with systems of equations as it allows you to work with simpler, single-variable equations. In our problem:

• Substitute \(f = g + 1\) from Equation 2 into Equation 1 to get a single equation with one variable: \(g = 3(g + 1) - 2\).
• After simplification, we get \(g = -\frac{1}{2}\).
• We then use this value to find \(f\) by substituting back into \(f = g + 1\): \(f = -\frac{1}{2} + 1\), yielding \(f = \frac{1}{2}\).

The substitution method reduces a system of equations to a simpler, single-variable equation, making it easier to solve for the unknowns step by step. Always check your substituted values in both original equations to ensure they satisfy the system as a whole.