Tag: teaching systems of linear equations

Posted in Algebra, Geogebra

Solving systems of linear equations by elimination method

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This short investigation  about the graphs of the sum and difference of two or more linear equations may be used as an introductory activity to the lesson on solving systems of linear equations by elimination. It will provide a visual explanation why the method of elimination works, why it’s ok to add and subtract the equations.

The  investigation may be introduced using the GeoGebra applet below.

1. Check the box to show the graph when equations b and c are added.

2. Where do you think will the graph of b – c pass? Check box to verify prediction.

3. Check the box to show graphs of the sum or difference of two equations. What do you notice about the lines? Can you explain this?
[iframe https://math4teaching.com/wp-content/uploads/2011/07/solving_systems_by_elimination.html 700 400]

When equations b and c intersect at A. The graph of their sum will also intersect point A.

b:  x + 2y =1

c:  x – y =-5

a:  2x+y=-4

After this you can then ask the students to think of a pair of equation that intersect at a point and then investigate graph of the sum and difference of these equations. It would be great if they have a graphing calculator or better a computer where they can use GeoGebra or similar software. In this investigation, the students will discover that the graphs of the sum and difference of two linear equations intersecting at (p,r) also pass through (p,r). Challenge the students to prove it algebraically.

If ax+by=c and dx+ey=f intersect at (p,r),

show that (a+d)x+(b+e)y=f +c also intersect the two lines at (p,r).

The proof is straightforward so my advise is not to give in to the temptation of doing it for the students. After all they’re the ones who should be learning how to prove. Just make sure that they understand that if a line passes through a point, then the coordinates of that points satisfies the equation of the line. That is if ax+by=c passes through (p,r), then ap+br=c.


The investigation should be extended to see the effect of multiplying the linear equation by a constant to the graph of the equation or to start with systems of equations which have no solution. Don’t forget to relate the results of these investigations when you introduce the method of solving systems of equation by elimination. Of course the ideal scenario is for students to come up with the method of solving systems by elimination after doing the investigations.

You can give Adding Equations for assessment.

Posted in Geogebra, Geometry

Geometric relations – angles made by transversal

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Geometry is a natural area of mathematics for which students should develop reasoning and justification skills and their appreciation of the logico-deductive part of mathematics that build across the grades. Learning tasks therefore should be so designed so that the focus of the learning is on the development of these skills as well and not merely on the learning of facts.

Consider the GeoGebra applets in Figures 1 and 2 below. Which of them will you use for teaching the relationships among the angles made by transversal with parallel lines? Before this lesson of course, the students already learned about linear pairs. Click the figures below to explore the applets before you continue reading.

In the first figure, dragging D or F along the parallel lines, the students will observe that there are angles that will always be equal. Thus from this, they can make the following conjectures:

(1) the alternate interior angles are equal;

(2) the vertical angles are equal;

(3) the corresponding angles (a pair of interior and exterior angles on the same side of the transversal) are equal; and,

(4) the pair of exterior and interior angles on the same side of the transversal sum up to 180 degrees.

In all these cases, the students are reasoning inductively. They will generalize from the measures they observed. Because of this, there seem to be no need for proof since there were bases for the generalizations. The measures of the angles. In this activity students will have learned geometric facts but not the geometric reasoning. Inductive reasoning maybe, but not deductive reasoning.

Contrast the first applet  to the second one. Dragging D or F along the parallel lines, the students will observe that the sum of the pair of exterior -interior angles on the same side of the transversal is always 180 degrees. They will also observe that the other angles also changes. The teacher can then challenge the students to make predictions about the measures of these angles and the relationships among them. These will create a need for proof.

And how should the proof look like? My suggestion is not to be very formal about it like using a two-column proof. For example, to prove that measures of vertical angles are always equal they can set up their proof like these:

To prove p = t:

p + s = 180

s + t = 180

p + s = s + t

p = t.

Students can very well set-up an explanation like this. They have seen it when they learned about solving systems of linear equation. What more, it uses the very important property of equality – the transitive property: If a = c, and b = c, then a = b. Great way to link algebra and geometry.

Posted in Algebra, Math investigations

Solving systems of equations by elimination – why it works

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Mathematical knowledge is only powerful to the extent to which it is understood conceptually, not just procedurally. For example, students are taught the three ways of solving a system of linear equation: by graphing, by substitution and by elimination. Of these three methods, graphing is the one that would easily make sense to many students. Substitution, which involves expressing the equations in terms of one of the variables and then equating them is based on the principle of transitive property: if a = c and b = c then a = b. But, what about the elimination method, what is the idea behind it? Why does it work?

While the elimination method seems to be the most efficient of the three methods especially for linear equations of the form ax + by = c, the principle behind it is not easily accessible to most students.

Example: Solve the system (1) 3x + y = 12 , (2) x – 2y = -2.

To solve the system by the method of elimination by eliminating y we multiply equation (1) by 2. This gives the equation (3), 6x + 2y = 24. Thus we have the resulting system,

6x + 2y = 24
x – 2y = -2.

The procedure for elimination tells us that we should add the two equations. This gives us a fourth equation (4), 7x = 22. We can then solve for x and then for y. But we have actually introduced 2 more equations, (3) and (4) in this process. Why is it ok to ‘mix’ these equations with the original equations in the system?

Equation (3) is easy to explain. Just graph 3x + y = 12 and 6x + 2y = 24. The graph of these two equations coincide which means they are equal. But what about equation (4), why is it correct to add to any of the equations? The figure below shows that equation (4) will intersect(1) and (2) at the same point.

Is this always the case? Think of any two linear equations A and B and then graph them. Take the sum or difference of A and B and graph the resulting equation C. What do you notice? This is the principle behind the procedure for the elimination method. But before students can do this investigation, they need to have some fluency on creating equation passing through a given point. The following problem can thus be given before introducing them to elimination method.

Is there a systematic way of generating other equations passing through (3,1)? This will lead to the discovery that when two linear equations A and B intersect at (p,q), A+B will also pass through (p,q). With little help, students can even discover the elimination method for solving systems of linear equations themselves from this. This problem is again another example of a task that can be used for teaching mathematics through problem solving . The task also links algebra and geometry. Click this link for a proposed introductory activity for teaching systems of equation by elimination method.