Problem 68 - College Algebra

Question

What is a system of linear equations? Provide an example with your description.

Step-by-Step Solution

Answer

A system of linear equations is a set of two or more linear equations, where each equation shares the same set of variables. An example would be the system: \[ \begin{align*} 2x + 3y &= 8 \\ x - y &= 1 \end{align*} \]

Step 1 - Define System of Linear Equations

A system of linear equations is a collection of two or more linear equations that all contain the same set of variables. A linear equation is one that can be expressed in the form \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants, and \( x \) and \( y \) are variables.

Step 2 - Explain The Variables

The variables \( x \) and \( y \) can represent different aspects based on the problem at hand. Solving the system of linear equations implies finding the values of these variables that make all the equations in the system true simultaneously.

Step 3 - Provide an Example

For instance, here is a simple system of two linear equations in two variables: \[ \begin{align*} 2x + 3y &= 8 \\ x - y &= 1 \end{align*} \]. This system represents two lines in a two-dimensional space, and the solution to the system represents the point of intersection of these two lines.