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.
it’s been a long time since i wrote this post. the letters a-k are labels and therefore not to be operated on. maybe i should think of another way of presenting the relationships.
yes, will correct the f+c part. it’s hard to correct the applet though. i’ll just make a new one. about adding/subtracting equations and function, it does throw my own intuition too:-) i’ll see what i can figure out when i get the time. thank you for pointing them out and for the corrections.
This is very nice. (I think you need c+f on the RHS of the ‘challenge’ though)
Do you see any interference in the students minds between the concepts of adding equations and adding functions? I know it sometimes throws my own intuition off and I wonder if you have any ways of helping kids get a feel for the difference.
Also I was puzzled that (b+c)-(b-c) didn’t seem to be giving 2c but I think your equation g (giving the solution for x) should have been labelled a+c (or b+2c), rather than a-d.