Solving systems of simultaneous equations graphically
A linear equation has infinite solutions. If we consider an equation of the form , there are an infinite amount of pairs that satisfy it. But, when we consider multiple equations and their common solutions, we could have one, infinite or none pairs of that are solutions of the equations at the same time.
A system of simultaneous equations with two unknowns can be solved graphically, as the equations correspond to lines.
If the system has a solution, it will manifest as the intersection point between both lines. The x and y coordinates of this point will be the two common solutions for the unknown numbers.
In the next applet, modify the equations to represent different systems. Try to find the solution for each pair of simultaneous equations. Then, answer the questions below.
Sometimes, it is not possible to find a common solution for both equations. What happens with the lines when this happens?
Write a system of simultaneous equations without a solution.
Other times, a system of simultaneous equations has infinite solutions. Write an example where this happens.
Beyond graphs
Most times, it is not confortable to use the graphing method to solve a system of simultaneous equations. When the common solutions are not whole numbers, it's hard to see the coordinates of the intersection point. Also, when solving without technology, graphing can take too much time.
Also, when dealing with more than two variables, 2D space is not enough. Each new variable you add is a new dimension you would need to to represent the system graphically, so it would be impossible to do with more than 3 variables.
Luckily, there are algebraic methods to solve systems of equations. Also, most calculators solve them automatically.