Explainer: Consistency and Dependency of Linear Systems

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: 𝑦=π‘šπ‘₯+𝑐,

where π‘š is the slope of the line and 𝑐 is the 𝑦-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 π‘₯+7𝑦=20 and 2π‘₯+14𝑦=40?

Answer

Here, you may notice directly that we can divide the second equation by 2, throughout, to get π‘₯+7𝑦=20.

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 𝑦-intercepts. If we subtract π‘₯ from each side of the first equation and divide by 7 throughout, we get 𝑦=βˆ’17π‘₯+207.

If we do the same for the second equation and subtract 2π‘₯ from both sides and divide by 14 throughout, we again get 𝑦=βˆ’17π‘₯+207.

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: 𝑦+π‘₯=3,π‘¦βˆ’2π‘₯=5.

Answer

If we rearrange each of the equations from the system into slope-intercept form (𝑦=π‘šπ‘₯+𝑐, where π‘š is the slope and 𝑐 is the 𝑦-intercept), we can easily identify the nature of the lines. For the first equation, if we subtract π‘₯ from both sides, we get 𝑦=βˆ’π‘₯+3, and for the second equation if we add 2π‘₯ to both sides, we get 𝑦=2π‘₯+5.

We can see now that the equations are in this form, that the two equations are not parallel (as one has a slope of βˆ’1 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: 3𝑦=9+3π‘₯,3𝑦+3π‘₯=15.

Answer

If we rearrange each of the equations from the system into slope-intercept form (𝑦=π‘šπ‘₯+𝑐, where π‘š is the slope and 𝑐 is the 𝑦-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 𝑦=βˆ’π‘₯+3, and for the second equation if we subtract 3π‘₯ from both sides and divide by 3 throughout, we get 𝑦=βˆ’π‘₯+5.

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 βˆ’1) but not coincident as the intercepts are different. This means that the system has no solutions, so the answer in this case is β€œnone.”

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 𝐿 and 𝐿 have an infinite number of solutions. Which of the following describes the relationship between the lines 𝐿 and 𝐿?

  1. They are coincident.
  2. They are parallel.
  3. They are perpendicular.
  4. They intersect in one point.

Answer

If we think about the nature of the lines 𝐿 and 𝐿, 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 6π‘₯+7𝑦=2 and 18π‘₯+π‘˜π‘¦=6 have an infinite number of solutions, what is the value of π‘˜?

Answer

With this question, we are asked to determine the value of π‘˜ 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 18π‘₯+21𝑦=6, which matches the coefficient of π‘₯ 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 π‘˜=21.

Key Points

  1. A system of equations has a single solution if the lines intersect (if their slopes are different).
  2. A system of equations has infinitely many solutions if the lines are in the same position (they are coincident).
  3. A system of equations has no solutions if the lines are parallel (and not coincident).
  4. 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.

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