5.1 Linear Equation Systems
An equation system is a collection of several equations (containing several unknowns). An equations system can be solved in different ways. We will show you here two of them.
Imagine that you have two equations with the unknown variables x and y. These two equations are included in an equation system.
Here is an example with two equations:
Equation 1: y = x + 2
Equation 2: y = 2x + 8
In order to show that these are included in the same equation system we usually write them together with a curly bracket:
We will now show how to solve these with both a graphical method and an algebraic one.
Graphical Solution
When we draw graphs for the two straight lines y = x + 2 and
y = 2x + 8 in a coordinate system, it looks like this:
We can then determine that the graphed lines intersect each other at the point (2, 4).
As a result, the equation system has the solution x = 2, y = 4.
Algebraic Solution
We will show an algebraic method called replacement. We are using the same example as above:
We are looking for an x and y value that solve both equations.
Suggestion: y = y and so must even x + 2 = 2x + 8
x + 2 = 2x + 8
3x = 6
x = 2
In order to find the yvalue, we then replace the xvalue with 2 in either of the equations:
y = x + 2
y = 2 + 2
y = 4
With this we see that the solution to the equation system is:
Sometimes you need to first isolate x or y on one side in one of the equations in the equation system:
1. Get x to one side in equation 2:
x + y = 1
x = 1 – y
2. Then replace x with x = 1  y in equation 1.
2y  x = 5 (1)
2y – (1 – y) = 5
2y – 1 + y = 5
3y = 6
y = 2
3. We then replace y with 2 in either of the equations:
x + y = 1 (2)
x + 2 = 1
x = 1
4. With this, we see that the solution to the equation system is:
