Systems of Equations

In mathematics, a system of equations is a collection of two or more equations involving the same set of unknown variables. The goal of solving a system of equations is to find values for the unknown variables that satisfy all the equations simultaneously.

Types of Systems of Equations

1. Linear Systems:

• In a linear system, all equations are linear, meaning they involve only linear combinations (multiplication by constants and addition) of the variables.
• For example, a system of linear equations could be: 2x+ 3y&= 10\\ 4x- y&= 3\end{align*}$$

2. Nonlinear Systems:

• In a nonlinear system, at least one equation is nonlinear, meaning it contains a term involving the product of variables or a variable raised to a power.
• An example of a nonlinear system could be: x^2+ y&= 5\\ x+ y^2&= 3\end{align*}$$

Solving Systems of Equations

1. Graphical Method:

• In this method, you graph each equation on the same set of axes and find the point(s) where the graphs intersect. These points represent the solutions to the system.

2. Substitution Method:

• In the substitution method, you solve one equation for one variable and substitute that expression into the other equation.

3. Elimination Method:

• In the elimination method, you add or subtract the equations to eliminate one of the variables, then solve for the remaining variable.

Real-life Example

Suppose you have $100tospendonbuyingapplesandoranges.Ifeachapplecosts$2 and each orange costs $1, and you want to buy a total of 60 fruits, how many apples and oranges can you buy?

Leta be the number of apples ando be the number of oranges. We can form a system of equations:$$ \begin{align*} 2a + o &= 100 \ a + o &= 60 \end{align*}

By solving this system of equations, you can determine that you can buy 40 apples and 20 oranges with$100. ## History The study of systems of equations dates back to ancient civilizations such as Babylonian mathematics, where methods for solving linear equations were developed. The concept of systems of equations has been fundamental in the development of algebra and its applications in various fields such as engineering, physics, and economics. ## Exam Questions 1. Solve the following linear system of equations: $$\begin{align*} 3x- 2y&= 7\\ 2x+ 5y&= 1\end{align*}$$ 2. Determine whether the following system of equations has a unique solution, no solution, or infinitely many solutions: $$\begin{align*} x+ 2y&= 5\\ 2x+ 4y&= 10\end{align*}$$ 3. Solve the following nonlinear system of equations: $$\begin{align*} x^2+ y&= 10\\ x+ y^2&= 7\end{align*}$$