Systems of equations in two variables
Find the solutions of a system with an equation of degree 2 as common points of graphs, solve it by substitution and form a system from a word problem.
- Germany: grade 10
- Palestine: grade 11
- Ukraine: grade 9
- Russia: grade 8
Guide
In Ukrainian Grade 9 algebra (§13–14) a solution of a system of equations in two variables is a pair of numbers (x,y) that makes every equation true. On the graphs it is a common point of the graphs of all the equations.
The graph of (x − a)2 + (y − b)2 = R2 is a circle with centre (a,b) and radius R, and the graph of y = ax2 + bx + c is a parabola. So a system can be solved graphically: draw both graphs and find where they cross.
For exact answers use substitution: express one variable from the equation of degree 1 and put it into the other equation. You get an equation in one variable.
Many word problems turn into a system of two equations: introduce two variables, write two relations between them, solve the system and check that the answer fits the problem.