Systems of Linear Equations

Geometric idea/Introduction

We want to know when a collection of lines has a point that lies on all of the lines. There are three possible situations: the lines intersect in a point, all of the equations describe one line, and the lines do not intersect at all. If there is at least one solution we say the system is consistent, otherwise it is inconsistent. The methods that are used are substitution and elimination.

Solving by Substitution

For solving by substitution, we solve for one variable in one of the equations, and then substitute it into the other equation. This results in an equation with one variable which we can solve. Then we back solve for the last variable

Example:

Solve: 2x + y = 5 and -4x + 6y = -2

In the first equation we can solve for y, seeing that y = 5 - 2x. Substituting that into the second equation we find that -4x + 6(5-2x) = -2. Solving for x, we find that x = 2. This means y = 1.

Solving by Elimination

In this method, we multiply each equation by constants so the coefficients of one of the variables are the same, or off by a factor of -1. Then subtracting (or adding) the equations together eliminates one of the variables.

Example:

Solve: 2x + y = 5 and -4x + 6y = -2.

Multiplying the first equation by 2, we get 4x + 2y = 10 and -4x + 6y = -2. Adding these two equations together, we get 8y = 8. so y = 1. Plugging that into the first equation we can see that x = 2.

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