The systems in this section will consist of two linear equations and two unknowns Given linear equations, we are asked to find out if they have simultaneous solutions. In other words, where do the two lines intersect? This question brings up three cases:

*x*,

*y*). The point where they cross is the solution to the system.

However, not all linear systems have on ordered pair solution; some have no common points and others have infinitely many. Imagine that you were asked to solve the system consisting of two parallel lines, where do they cross? In this case, there is no simultaneous solution and the system of two parallel lines is inconsistent. In the case where the system consists of two lines that happen to be the same line, there are infinitely many common points. This system is dependent and solutions can be presented in the form (

*x*,*y*) where*x*can be any real number and*y*=*mx*+*b*.**Solve the systems given the graph**:

**Is the ordered pair a solution to the system?**

**Solve the system**:

**Step 1**: Place the linear equations in slope-intercept form.

**Step 2**: Graph the lines and use the graph to find the common point.

**Step 3**: Check your answer and present it as an ordered pair.

**Solve the system using the graphing method**:

**Solve the system using the graphing method**:

**Solve the system using the graphing method**:

**Solve the system using the graphing method**:

Dependent systems seem to give beginning algebra students the most trouble. Remember that we are looking for points where the two lines intersect. If the lines are the same, well then they will cross at infinitely many points. Because of this we have to use special notation to indicate an infinite set. Notice that we have already put the line in

You might see different notation in other texts such as,*y*-intercept form,*y*=*mx*+*b*, so it is not a big leap to write the set final ordered pair solutions in the form (*x*,*mx*+*b*).

**Video Examples on YouTube**: