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One way to obtain such an ordered pair is by graphing the two equations on the same set of axes and determining the coordinates of the point where they intersect. The point of intersection is 3, 2. Thus, 3, 2 should satisfy each equation. We will develop methods for exact solutions in later sections.
Linear equations considered together in this fashion are said to form a system of equations. As in the above example, the solution of a system of linear equations can be a single ordered pair.
The components of this ordered pair satisfy each of the two equations. Some systems have no solutions, while others have an infinite number of solu- tions.
If the graphs of the equations in a system do not intersect-that is, if the lines are parallel see Figure 8. If the graphs of the equations are the same line see Figure 8.
Notice that when a system is inconsistent, the slopes of the lines are the same but the y-intercepts are different. When a system is dependent, the slopes and y-intercepts are the same. In our work we will be primarily interested in systems that have one and only one solution and that are said to be consistent and independent.
The graph of such a system is shown in the solution of Example 1. What is more, the solutions we obtain by algebraic methods are exact. The system in the following example is the system we considered in Section 8.Algebra Calculator - get free step-by-step solutions for your algebra math problems.
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