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The Elimination Method is one of the techniques used to solve systems of equations. This method involves eliminating one variable by adding or subtracting the equations.

In our example, we had two equations:
\( y = \frac{3}{4} x - 4\) and \( 4 y = 2 x + 3 \).

First, we need to rewrite both equations in standard form. For the first equation, this means eliminating the fraction by multiplying every term by 4, transforming it from \( y = \frac{3}{4} x - 4 \) to \( 4 y = 3 x - 16 \). The second equation, \( 4 y = 2 x + 3 \), is already in standard form.

Next, we align the equations and subtract the second equation from the first to eliminate the variable \( y \). This gives us \( 0 = x - 19 \), leading to \( x = 19 \).

By eliminating one variable, we simplify the system and can easily solve for the remaining variable.

The Substitution Method involves solving one equation for one variable and then substituting that solution into the other equation.

In this context, we first solve one of the equations for one of the variables. For example, equation G is solved for \( y \)
\( y = \frac{3}{4}x - 4 \).

We then substitute \( x = 19 \) from our elimination result into this equation:
\( y = \frac{3}{4}(19) - 4 \).
This step-by-step method allows us to find the value of \(y\) as 10.25.

Substitution is particularly useful when one variable's coefficient is 1 or -1 because it simplifies the calculations.

Equations in Standard Form are presented as \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants and \( x \) and \( y \) are variables.

Rewriting equations in standard form makes it easier to apply the elimination method.

For example, transforming equation G from \( y = \frac{3}{4} x - 4 \) to \( 4 y = 3 x - 16 \) puts both equations on equal terms. This step simplifies the elimination process.

Equations in standard form allow for systematic manipulations, creating consistency across different problems and methods.