Systems of Equations and Solutions

SYSTEMS OF EQUATIONS AND SOLUTIONS

SYSTEM OF EQUATIONS

A “SYSTEM” is a set of equations with the same variables.

EX:

x - 3y = 12

2x + 4y = 4

We’re looking at systems of linear equations—equations of lines.

SOLUTIONS

The SOLUTION of a system of equations is made up of solutions that work for both equations.

EX: The solution to the equations above is x = 6, y = -2) because that point fits into both equations:

Plugging in 6 for x and -2 for y in the first equation, we get

6 – 3(-2) = 12

6 + 6 = 12

12 = 12

Plugging in 6 for x and -2 for y in the second equation, we get

2(6) + 4(-2) = 4

12 – 8 = 4

4 = 4

GRAPHING SYSTEMS

We can visualize systems by graphing the equations on the same set of axes. The solution is where the lines cross.

This is the graph of the example above. You can see the lines cross at (6, -2).

A system of equations may have ONE solution, MANY solutions, or NO SOLUTIONS.

If the lines cross ONCE, there is ONE specific solution.

If the lines are the SAME, they are crossing many, many times (at every single point on the line), so there are infinitely MANY SOLUTIONS.

If the lines are PARALLEL, there are NO SOLUTIONS, because the lines never cross.

(Note that these lines have the same slope but different intercepts.)