In this explainer, we will learn how to determine the number of solutions for a system of linear equations and whether each system is consistent, inconsistent, or dependent.
When we solve a system of two linear equations, we are looking to find the intersection between the two lines. We can do this using algebraic methods or graphically.
In order to determine the number of solutions to a system of linear equations, we can consider the graphs of each of the equations. Effectively, for a system of two equations, there are three possible scenarios: the system has no solutions, it has infinitely many solutions, or it has a unique solution. You can see these three cases here.
If the two lines are parallel, the system will have no solutions; if the two lines are coincident (in the same place), they will have infinitely many solutions; and in all other cases the system will have a unique solution.
If you are not given the graphs of the equations, however, it is not necessarily the most efficient method to draw the graphs of the two lines and then use this to determine the number of solutions. We can, in fact, determine the number of solutions directly from the equations of the system using algebraic methods.
Firstly, let us recap some important information that we can obtain from the equation of a straight line graph. The equation of a straight line graph can always be written in slope-intercept form:
where is the slope of the line and is the -intercept. This means that we can determine whether two lines are coincident, parallel, or intersecting by rearranging them into slope-intercept form. If the slopes and intercepts are equal, then the two lines are coincident; if just the slopes are equal, then they are parallel; and if the slopes are different, then the lines will intersect.
Let us have a look at some examples where we use the equations of the lines to determine the number of solutions to the system.
Example 1: Finding the Number of Solutions to a System of Equations
How many solutions are there to the simultaneous equations and ?
Answer
Here, you may notice directly that we can divide the second equation by 2, throughout, to get
This means that the two equations are coincident and, therefore, the system has infinitely many solutions. Alternatively, we could have taken the approach to rearrange the two equations into slope-intercept form so we can directly compare their slopes and -intercepts. If we subtract from each side of the first equation and divide by 7 throughout, we get
If we do the same for the second equation and subtract from both sides and divide by 14 throughout, we again get
This means that the two equations are coincident and, therefore, the system has infinitely many solutions.
Example 2: Identifying Information about a System of Equations given Information about the Number of Solutions
Find the number of solutions to the following system of equations:
Answer
If we rearrange each of the equations from the system into slope-intercept form (, where is the slope and is the -intercept), we can easily identify the nature of the lines. For the first equation, if we subtract from both sides, we get and for the second equation if we add to both sides, we get
We can see now that the equations are in this form, that the two equations are not parallel (as one has a slope of and the other has a slope of 2), and that, therefore, they must intersect. This means that the system has one unique solution.
Example 3: Finding the Number of Solutions to a System of Equations
Find the number of solutions to the following system of equations:
Answer
If we rearrange each of the equations from the system into slope-intercept form (, where is the slope and is the -intercept), we can easily identify the nature of the lines. For the first equation, if we divide the equation by 3 throughout and then change the order of the terms on the right-hand side, we get and for the second equation if we subtract from both sides and divide by 3 throughout, we get
We can see now that the equations are in this form and that the two equations are parallel (as they both have a slope of ) but not coincident as the intercepts are different. This means that the system has no solutions, so the answer in this case is “none.”
With some questions, you may be asked to use your knowledge of the conditions on a system of equations to determine or dictate the number of solutions. Let us look at a couple of examples where we have to apply our understanding of equations to answer the question.
Example 4: Determining the Geometry of a System of Equations
The simultaneous equations represented by the straight lines and have an infinite number of solutions. Which of the following describes the relationship between the lines and ?
- They are coincident.
- They are parallel.
- They are perpendicular.
- They intersect in one point.
Answer
If we think about the nature of the lines and , if the system has infinitely many solutions, this must mean that the lines are in the same position and, therefore, they are coincident. If this were not the case, either the lines would intersect and have a single solution, or they would be parallel which would mean the system would have no solutions.
Example 5: Finding the Number of Solutions to a System of Equations
If the simultaneous equations and have an infinite number of solutions, what is the value of ?
Answer
With this question, we are asked to determine the value of such that the equation has an infinite number of solutions. For this to be the case, the two lines must be coincident, that is, have the same equation. The simplest way to approach this problem is to look at the first equation (as we know all of the coefficients) and determine the number by which we need to multiply this to match the coefficients of the second equation. In this case, if we multiply the first equation by 3 throughout, we get which matches the coefficient of and the constant term of the second equation. If we then compare the two equations, for the lines to be coincident, their equations must be equal, so .
Key Points
- A system of equations has a single solution if the lines intersect (if their slopes are different).
- A system of equations has infinitely many solutions if the lines are in the same position (they are coincident).
- A system of equations has no solutions if the lines are parallel (and not coincident).
- In order to determine the geometry of a system of equations, we can calculate the slopes and intercepts of the lines and compare them or consider the system graphically.
