Systems of Equations in Two Variables

Introduction

A system of equations consists of two or more equations made up of two or more variables such that all equations in the system are considered simultaneously. To find the unique solution to a system of equations, we must find a numerical value for each variable in the system that will satisfy all equations in the system at the same time. Some systems may not have a solution (for example, two distinct lines that are parallel to one another) and others may have an infinite number of solutions (for example, $y=0$ and $y=sin(x)$, since $sin(x)=0$ for an infinite number of x-values). In order for a linear system to have a unique solution, there must be at least as many equations as there are variables. Even so, this does not guarantee a unique solution.

Systems of equations can be solved in a number of ways. We can use elimination and/or substitution to solve for points of intersection shared by all equations in the system. We can also sometimes graph the equations to find the points shared by all equations, given that our system is graphable.

Here are some examples of systems of equations with two variables:

• One solution: $y=2x$ and $y=x+5$. By use of substitution, elimination, or graphing, we can find that the solution to this system is $(5,10)$, as this is the only point shared by both equations in the system.
• Multiple solutions: $y=x^{2}$ and $y=4$. Since $x^{2}=4$ when x = 2 or -2, we know there are two solutions to this system: (2, 4) and (-2, 4). This is also clear when we graph these two equations, as $y=4$ intersects the parabola $y=x^{2}$ when x = -2, and when x = 2.
• No solutions: $y=x^{2}$ and $y=-3$. Since $x^{2}$ is positive for all values of x, it cannot equal -3. When graphed, we can see that these two equations never intersect. Thus there are no solutions to this system of equations.
• Infinite solutions: $y=sin(x)$ and $y=0$. $sin(x)=0$ for all $x=\pi k$, where k is some integer. So, this system has an infinite number of solutions of the form $(\pi k,0)$; that is, the solution set for this system of equations is {.....$(-2\pi ,0)$, $(-\pi ,0)$, $(0,0)$, $(\pi ,0)$, $(2\pi ,0)$.....}.

Linear Equations with Two Variables

In the previous module, linear equations with two variables were discussed. A single linear equation having two unknown variables is practically insufficient to solve or even narrow down the solutions for the two variables, although it does establish a relationship between them. The relationship is shown graphically as a line. Another linear equation with the same two variables may be enough to narrow down the solution to the two equations to one value for the first variable and one value for the second variable, i. e. to solve the system of two simultaneous linear equations. Let's see how two linear equations with the same two unknowns might be related to each other. Since we said it was given that both equations were linear, the graphs of both equations would be lines in the same two-dimensional coordinate plane (for a system with two variables). The lines could be related to each other in the following three ways:

1. The graphs of both equations could coincide giving the same line. This means that the two equations are providing the same information about how the variables are related to each other. The two equations are basically the same, perhaps just different versions or forms of each other. Either one could be mathematically manipulated to produce the other one. Both lines would have the same slope and the same y-intercept. Such equations are considered dependent on each other. Since no new information is provided, the addition of the second equation does not solve the problem by narrowing the solution set down to one solution.

Example: Dependent linear equations

$6x-3y=12\$ $y=2x-4\$

The above two equations provide the same information and result is the same graph, i. e. lines which coincide as shown in the following image.

Let's see how these equations can be mathematically manipulated to show they are basically the same.

Divide both sides of the first equation $6x-3y=12\$ by 3 to give

$2x-y=4\$

Now add y to both sides

$2x=4+y\$

Now subtract 4 from both sides

$y=2x-4\$

This is the same as the second equation in the example. This is the slope-intercept form of the equation, from which a slope and a y-intercept unique to the line can be compared with any other equations in the slope-intercept form.

2. The graphs of two lines could be parallel although not the same. The two lines do not intersect each other at any point. This means there is no solution which satisfies both equations simultaneously, i. e. at the same time. The solution set for this system of simultaneous linear equations is the empty set. Such equations are considered inconsistent with each other and actually give contradictory information if it is claimed they are both true at the same time in the same problem. The parallel lines have equal slopes but different y-intercepts.

Sets of equations which have at least one common point which might provide a solution set are consistent with each other. For example, the dependent equations mentioned previously are consistent with each other.

Example: Inconsistent linear equations

$3x-2y=-2\$ $3x-2y=2\$

To compare slopes and y-intercepts for these two linear equations, we place them in the slope-intercept forms. Subtract 3x from both sides of both equations.

$3x-2y=-2\qquad \qquad 3x-2y=2$ $-2y=-3x-2\qquad \qquad -2y=-3x+2$

Divide both sides of both equations by -2 and simplify to get slope-intercept forms for comparison.

$(-2y)/(-2)=(-3x-2)/(-2)\qquad (-2y)/(-2)=(-3x+2)/(-2)$

$y={\frac {3}{2}}x+1\qquad \qquad \qquad \qquad \qquad y={\frac {3}{2}}x-1$

Now, both slopes are equal at 3/2, but the y-intercepts at 1 and -1 are different.
The lines are parallel. The graphs are shown here:

3. If the two lines are not the same and are not parallel, then they would intersect at one point because they are graphed in the same two-dimensional coordinate plane. The one point of intersection is the ordered pair of numbers which is the solution to the system of two linear equations and two unknowns. The two equations provide enough information to solve the problem and further equations are not needed. Such equations intersecting at a point providing a solution to the problem are considered independent of each other. The lines have different slopes but may or may not have the same y-intercept. Because such equations provide at least one solution point, they are consistent with each other.

Example: Consistent independent linear equations

$y=3x-5\$ $y=-x-1\$

Both of these equations are given in the slope-intercept, so it is easy to compare slopes and y-intercepts. For these two linear functions, both slopes are different and both y-intercepts are different. This means the lines are neither dependent nor inconsistent, so on a two-dimensional graph they must intersect at some point. In fact, the graph shows the lines intersecting at (1,-2), which is the ordered pair solution to this system of independent simultaneous equations. Visual inspection of a graph cannot be relied on to give perfectly accurate coordinates every time, so either the point is tested with both equations or one of the following two methods is used to determine accurate coordinates for the intersection point.

Resources

• Linear Systems with Two Variables, Paul's Online Notes
• Systems of Linear Equations with Two Variables, OpenStax
• Systems of Nonlinear Equations with Two Variables, OpenStax
• Solving Systems of Equations with Elimination, patrickJMT
• Solving Systems of Equations with Substitution, patrickJMT
• Solving Systems of Equations with Graphing, patrickJMT

Licensing

Content obtained and/or adapted from:

• Systems of Linear Equations: Two Variables, OpenStax: College Algebra under a CC BY license
• Systems of Equations, Wikibooks: Algebra under a CC BY-SA license