Problem 85 - College Algebra Essentials

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 collection of two or more linear equations involving the same set of variables. A solution to a system of linear equations is a set of values that satisfy all the equations in the system simultaneously. This can be represented graphically as a point where the lines represented by each equation intersect. For example, a system of linear equations is given by \[3x+2y-1=0\] and \[2x-3y+1=0\], and its solution would be the values of \(x\) and \(y\) that satisfy both equations.

Step 1: Definition of a System of Linear Equations

A system of linear equations is a collection of two or more linear equations involving the same set of variables. For example, consider the following system of linear equations: \[3x+2y-1=0\] \[2x-3y+1=0\] Here, both equations are linear (degree one), and they have the same variables, x and y.

Step 2: Describe a Solution to the System

A solution to a system of linear equations is a list of numbers that makes each equation in the system true. In other words, a solution is the set of values for the variables that satisfy all the equations in the system simultaneously. For the system given as an example, the solution would be the values of \(x\) and \(y\) that satisfy both equations.

Step 3: Graphical Representation

Each equation in a system of linear equations can be represented graphically as a line on a two-dimensional coordinate plane. A solution to the system, as described in the previous step, is represented as the point where these lines intersect. So, if we were to graphically represent the example system, we would graph the lines represented by both equations and identify the point of intersection as the system's solution.