Problem 85 - Algebra and Trigonometry

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. For example, the equations \(x + y = 5\) and \(2x - 3y = -7\) represent a system of two linear equations in two variables, x and y. The solution to this system, \(x=3, y=2\), is the set of values that satisfy all the equations in the system simultaneously.

Step 1: Defining a System of Linear Equations

A system of linear equations is a collection of two or more linear equations, each having the same variables. By 'linear', it is implied that all equations in the system are represented graphically as straight lines. The system is composed of equations structured such that the highest power of the variables is one.

Step 2: Explaining a Solution

A solution to a system of linear equations is an ordered pair, (or ordered triple, quadruple, etc., depending on the number of variables), that, when substituted into each equation, makes each equation true.

Step 3: Providing an Example

As an example, consider the system of two linear equations:\[ x + y = 5\]\[2x - 3y = -7\]These two equations form a system of linear equations. A solution to this system is a pair of values (x, y) that makes both equations true when substituted. The solution to this particular system is \(x=3, y=2\)