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A rational expression is like a fraction you might use in everyday life. It consists of numerators and denominators, but instead of just numbers, both the numerator and the denominator are polynomials. In simpler terms, these are algebraic fractions where the top and bottom are made of expressions involving variables. Understanding rational expressions helps in simplifying complex fractions and solving equations involving fractions.

They're key in algebra because they allow us to express a wide range of problems. However, these expressions require specific methods for simplification and calculation, such as partial fraction decomposition. In this particular exercise, the rational expression is \(\frac{3x + 50}{(x-9)(x+2)}\). Learning how to break them down into simpler parts can make them much easier to manage and solve.

A system of equations involves solving multiple equations together that share two or more variables. In algebra, it's common to find yourself working with these systems, especially when trying to determine unknown constants or variables from a given problem.

Systems can be solved using various methods like substitution, elimination, or graphical methods. In this exercise, a system of equations is derived from equating the numerators. Here, we have two equations from the partial fraction decomposition: \(A + B = 3\) and \(2A - 9B = 50\). Solving these reveals the specific values of A and B needed for decomposition.

These simultaneous equations help break down complex rational expressions into simpler, easier to work with components.

The substitution method is a straightforward way to solve systems of linear equations. It involves expressing one variable in terms of another and then substituting that expression into another equation. This method is particularly useful when one equation in the system is easy to manipulate.

In this exercise, we solved the system by substitution. Starting with the equation \(A + B = 3\), we expressed B as \(B = 3 - A\). This expression for B was then substituted into the second equation \(2A - 9B = 50\). The substitution method simplifies solving for A and B by reducing the number of unknowns at each step.

• Write one variable as an equation of the other. (e.g., \(B = 3 - A\))
• Substitute this expression in the other equation.
• Solve the resulting equation for the remaining variable.

By substituting and simplifying, we found that \(A = 7\) and \(B = -4\), leading us to complete the partial fraction decomposition step.

College Algebra focuses on building foundational skills to manage complex equations, functions, and expressions in mathematics. This includes conducting operations on rational expressions, solving systems of equations, and various methods like substitution.

Understanding these topics is crucial as they're regularly applied in calculus, engineering, and in solving real-life problems. The partial fraction decomposition shown in this exercise is just one example of how algebraic techniques simplify expressions.

• Develop skills to manipulate algebraic expressions.
• Use logical reasoning to solve equations.
• Apply methods like substitution and elimination effectively.

By mastering these concepts, students are better prepared for various mathematical challenges and equipped to apply these techniques in a variety of contexts beyond pure mathematics.