91Ó°ÊÓ

Linear equations are foundational in algebra and represent the simplest form of equations. They are called 'linear' because their graph results in a straight line when plotted on a coordinate plane. In the context of algebra, a linear equation typically includes variables raised only to the first power.

A general form of a linear equation can be expressed as:

• \( ax + by = c \)

Where \(a\), \(b\), and \(c\) are constants, and \(x\) and \(y\) are variables.

The goal when working with linear equations is often to solve for one variable in terms of the other, which was our first step in the exercise where we rearranged the equation to get \(y = 4x - 100\). This manipulation makes it easier to perform substitutions in systems of equations. Linear equations form the building blocks for more complex mathematical concepts.

A system of equations consists of two or more equations with a shared set of unknowns. Solving a system of equations involves finding values for the variables that satisfy all equations simultaneously.

There are several methods to solve systems of equations, and the substitution method is one of them. This method is particularly useful when one equation is already solved for one variable or can be easily rearranged. In the provided exercise, we used substitution by expressing \(y\) in terms of \(x\) from the first equation and then substituting this expression into the second equation.

Solving systems of equations can lead to three possible outcomes:

• A single solution (the lines intersect at one point).
• No solution (the lines are parallel and never intersect).
• Infinite solutions (the lines lie on top of one another).

In our case, the solution was a single point, illustrating that the two equations intersected at \((3800, 15100)\). Understanding the relationships between the equations is essential for interpreting the results correctly.

Set notation is a way to represent a collection of elements, typically numbers that satisfy a particular condition. It is often used to express the solution of systems of equations, as it succinctly captures all possible solutions.

In mathematics, a set is written with curly braces, like this: \(\{ \cdot \}\). For a solved system of equations, the solution is usually represented as a set containing ordered pairs, each pair representing the coordinate that satisfies all equations in the system.

In our example, the solution to the system of equations was expressed in set notation as \(\{(3800, 15100)\}\). This indicates that \(x = 3800\) and \(y = 15100\) is the sole solution to our system. Using set notation makes it easy to communicate all solutions efficiently, even in more complex cases with multiple solutions.