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Simplifying equations can make solving them much easier. Simplification involves reducing the complexity of a given problem by eliminating common factors or reducing equations to their simplest form. Whenever presented with an equation, always look for ways to simplify it: - Check if all terms share a common factor. - Divide the entire equation by this factor to make calculations easier and quicker. In the provided exercise, each term in the equations was divisible by 25. This allowed us to transform the original system:\[ \begin{align*} 250R + 225Z &= 400 \ 375R - 675Z &= 325 \end{align*} \]into:\[ \begin{align*} 10R + 9Z &= 16 \ 15R - 27Z &= 13 \end{align*} \]This process not only simplifies arithmetic operations but makes it far more conducive to applying algebraic methods like the elimination method.

Elimination is one of the primary methods of solving systems of equations. The goal is to get rid of one of the variables, making it easier to solve for the other. Here is a simple way to apply the method:- First, manipulate the equations such that adding or subtracting them will eliminate one of the variables. - This involves aligning coefficients to be equal (but opposite in sign) by multiplying through by appropriate numbers.In this exercise, the first equation was multiplied by 3 to align the coefficient of \( Z \) with that in the second equation:\[ 30R + 27Z = 48 \]Now, both equations share the same coefficient for \( Z \) (i.e., \( 27 \)). By subtracting, the \( Z \) terms cancel out, leaving a simpler equation in just \( R \) to solve. This method is powerful and can be applied to any linear system with skill and practice.

Recognizing and using common factors is a fundamental algebraic method that simplifies work and avoids errors. Common factors refer to numbers or variables that are shared by terms in an equation and can be used to factorise and simplify the equation.- Identify if there is any number or variable that divides every term in the equation evenly. - For this exercise, 25 divides each term in the initial equations. - Use this common factor to simplify the equations by division, leading to easier numbers to work with:\[ \begin{align*} \frac{250R}{25} + \frac{225Z}{25} &= \frac{400}{25} \ \frac{375R}{25} - \frac{675Z}{25} &= \frac{325}{25} \end{align*} \]Simplifying using common factors reduces the size and complexity of the system, making subsequent steps more straightforward and manageable.

Systems of equations involve finding values that satisfy multiple equations at the same time. When dealing with two equations with two variables, such as \( R \) and \( Z \), we need techniques like elimination or substitution:- Start by simplifying equations using any common factors.- Choose an appropriate method (elimination or substitution) to find one variable first.- Substitute back to find the other variable.In this exercise, the elimination method was chosen after simplifying the equations. With \( R \) found to be \( \frac{7}{9} \), substituting back into one of the simplified equations allows us to find \( Z \): \[ \begin{align*} 10\left(\frac{7}{9}\right) + 9Z &= 16 \end{align*} \] Concluding with \( Z = \frac{74}{81} \). Solving systems of equations is about methodical application of algebraic rules and are key in modeling and solving real-world problems.