Systems of Linear Equations

Not infrequently, the best way to model a real world situation is by using two or more variables. When the model is made entirely of linear equations, then we can use the methods in this chapter for solving the system of linear equations, assuming solution(s) exist. By far the easiest version of these systems is a system with two variables and two linear equations. 

Linear Equations

The definition of a linear equation given in can be extended to more variables. Any equation of the form a₁x₁+a₂x₂+...+a_nx_n = b, for real numbers a₁, a₂, ..., a_n (all nonzero) and b, is a linear equation or a first-degree equation in n unknowns.

System of Equations

A system of equations is a set of two or more equations, in two or more variables. for which a common solution is sought.

A set of equations considered simultaneously is called a system of equations. The solutions of a system of equations must satisfy every equation in the system. If all the equations in a system are linear, the system is a system of linear equations, or a linear system.

Solving Systems Graphically

A system of two linear equations with two variables can be graphed on the same coordinate axes. In this case, any solution(s) to the system will be points where both of the equations are true simultaneously, where the two lines intersect or overlap.

 There are three possible outcomes from a system of two variables. Here is an illustration from the text of the three scenarios:

Let's consider each of these three possibilities separately:

The Graphs of the Two Lines Intersect at One Point

This is by far the most common alternative in any algebra class. In this case, there is exactly one solution, and it is a point given by the coordinates, (x,y). Because there are one or more solutions the system is called consistent, and because neither equation is a multiple of the other one, the equations are said to be independent.

Example

Consider the system of linear equations below:

$y-x=1$

$y+x=3$

We can use the intercept method to graph these two equations since they are both in standard form.

Let $x=0\:$to find the y-intercept in the first equation. 

$y-x=y-\left(0\right)=1$

$y=1$

So, the y-intercept is $\left(0,1\right).$

Let $y=0\:$to find the x-intercept in the first equation,

$y-x=\left(0\right)-x=1$

$-x=1$

$x=-1$

So, the x-intercept is $\left(-1,0\right).$

Similarly, we can get the y-intercept and x-intercept for the second equation, which are $\left(0,3\right)\:$and $\left(3,0\right).$

The graph looks like this when we put both lines on the same coordinate axes:

Notice from the image that the two lines intersect at one point, the point $\left(1,2\right),$thus this system has exactly one solution, which is the point $\left(1,2\right).$

The Graphs of the Two Lines Are Parallel

In this case the two lines never intersect. Because the lines are parallel, the lines will have identical slopes but they will have different y-intercepts. This type of system of equations is called inconsistent because it has no solutions, and because neither equation is a multiple of the other one, the equations are said to be independent.

Example

Consider the system of linear equations below:

$y=-3x+5$

$y=-3x-2$

In this case, both equations are given in the slope-intercept form. Both of the equations have a slope on $-3\:$because the number multiplying the x variable is always the slope for any line in slope-intercept $\left(y=mx+b\right)\:$form.  The y-intercepts are the points $\left(0,5\right)\:$and $\left(0,-2\right),\:$so the graph of these two lines looks like this:

Since the two lines never intersect, this system of equations has no solution.

The Two Equations Have the Same Graph

Sometimes the two equations actually represent the identical line. In this case the lines represented by the two equations intersect at every point on the graph of the line represented by either of the equations. Thus, there are an infinite number of solutions, and the solutions are all points on the graph of either of the equations in the system. This type of system is consistent because there are solutions. However, the equations are said to be dependent because one equation is a multiple of the other.

Example

Consider the system of linear equations below:

$3y-2x=6$

$-12y+8x=-24$

We can see that one way to get the second equation would be to multiply both sides of the first equation by -4. If we divide the second equation by -4, then graph both equations using the intercept method, we get the following graph:

Since all points on this line are in the solution set of the system of equations, we write the solution using set-builder notation by choosing either of the equations so that the equation for the solution of this system would be $\lbrace\left(x,y\right)\mid\left(x,y\right)\:is\:on\:the\:line\:represented\:by\:3y-2x=6\rbrace.$

Note: It would also have been correct to replace the equation in the solution above with the other equation from the system, since all of the points are the same for either equation.

Examples

Here are some more examples of solving systems of linear equations by graphing:

On the next page, complete Homework 3.1.