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# Systems of Linear Equations

# Systems of Equations

A **system of equations** is a group of two or more equations. What, you were expecting robots?

### Sample Problem

The two equations

form a system of equations.

### Another Sample Problem

The three equations

also form a system of equations. A bigger one, although not necessarily better.

A **solution** to a system of equations is a selection of numbers for the variables that satisfy all the equations in the system at the same time. You are looking for a sort of skeleton key, a one-size-fits-all super-solution.

### Sample Problem

The values *x* = -15, *y* = -40 are a solution to the system of equations

First of all, these values satisfy the first equation. When *x* = -15 and *y* = -40, the left-hand side of the equation is -40. The right-hand side is

3(-15) + 5 = -45 + 5 = -40.

These values also satisfy the second equation. When *x* = -15 and *y* = -40, the left-hand side of the second equation is -40. The right-hand side is

2(-15) – 10 = -30 – 10 = -40.

Fortunately, -40 equals itself, so the values we were given pan out, and we can see that they are indeed a solution to the system of equations. Unfortunately, they are not the solution to all of life's problems. Would have been nice if they were.

Meanwhile, the values *x* = 1, *y* = 5 are *not* a solution to the system of equations

These values do satisfy the first equation. When *x* = 1 and *y* = 5, the left-hand side of the first equation is 5. The right-hand side is

3(1) + 2 = 5.

However, these values do *not* satisfy the second equation. When *x* = 1 and *y* = 5, the left-hand side of the second equation is 5. The right-hand side is

4(1) – 1 = 3.

3 does not equal 5, which is a good thing, since it would make counting strikes in baseball tricky. Since these values don't satisfy both equations at once, they are not a solution to the system of equations. Thanks for playing, kids.