Two Linear Equations in Two Variables

Suppose you want to solve a system of two linear equations in two variables. I will discuss elimination method. So, you have following general system: ax+by=c dx+ey=f Elimination method works as follows: express x (or y) from first expression and plug result into second. I will express x from first: x=(c-by)/a Now, plug result into second: d(c-by)/a+ey=f dc/a-by/a+ey=f y(e-b/a)=f-dc/a y=(f-dc/a)/(e-b/a) Now, x=(c-by)/a=(c-b(f-dc/a)/(e-b/a))/a A bit messy, right? But on practice it is more clear. Example 1. Solve the system 3x+2y=12 4x+5y=23 Solution. Express x from first equation: x=(12-2y)/3 Now, plug this result into second: 4*(12-2y)/3+5y=23 Multiply both sides of equation by 3: 4(12-2y)+15y=69 48-8y+15y=69 7y+48=69 7y=21 y=3 Now, x=(12-2y)/3=(12-2*3)/3=2 Thus, x=2 and y=3 Example 2. Solve the system x-4y=-3 y-3x=-2 Solution. Actually it doesn't matter from which equation to express variable and what variable to express. Let's express y from second equation: y=3x-2 Plug this result into first: x-4(3x-2)=-3 x-12x+8=-3 -11x=-11 x=1 Finally, y=3x-2=3*1-2=1 So, x=1 and y=1. Probably, you know that system of two linear equations in two variables has either one solution or no solution or infinitely many solutions. Let's see how elimination method works when we have two special cases: no solution or infinitely many solutions. Example 3. Solve the system 2x+3y=2 4x+6y=4 Solution. From first equation x=(2-3y)/2 Plugging this in first yields: 4(2-3y)/2+6y=4 2(2-3y)+6y=4 4-6y+6y=4 4=4 Wow! All variables have canceled out. Since 4=4 is correct equality then this system has infinitely many solutions. Example 4. Solve the system 2x+3y=1 4x+6y=4 Solution. From first equation x=(1-3y)/2 Plugging this in first yields: 4(1-3y)/2+6y=4 2(1-3y)+6y=4 2-6y+6y=4 2=4 Again all variables have canceled out. But, since 2=4 is incorrect equality then this system has no solution. Geometrically all above three cases have following meaning: System has one solution: lines intersect at one point. System has no solution: lines are parallel and don't coincide. System has infinitely many solutions: lines coincide.