07-1+Graphing+Systems+of+Equations

=7-1 Graphing Systems of Equations =

Two equations together are called a **system of equations**. A solution of a system of equations is an **ordered pair** of numbers that satisfies both equations. A system of two linear equations can have 0, 1 or an infinite number of solutions.


 * If the graphs intersect or coincide, the system of equations is said to be **consistent**. That is, it has //at least// one ordered pair that satisfies both equations.
 * If the graphs are parallel, the system of equations is said to be **inconsistent**. There are //no// ordered pairs that satisfy both equations.
 * Consistent equations can be **independent** or **dependent**. If a system has exactly one solution, it is independent. If the system has an infinite number of solutions, it is dependent.



**
 * Example 1: Use the graph below to determine whether each system has no solution, one solution , or infinitely many solutions.


 * 1) y = 2x + 3 and y = 2x - 4
 * 2) 6x - 3y = 12 and y = 2x - 4
 * 3) 6x - 3y = 12 and y = -x + 1


 * Answers:**
 * 1) This system has **no solution** because the lines are parallel, which means they will never intersect. I know that the lines are parallel because they have the same slope.
 * 2) This system has **infinitely many solutions** because the lines are the same line. I know that the lines are the same because I can put them both into slope intercept form and see that they are identical.
 * 3) This system has **one solution** because the lines intersect. Any two lines that have different slopes are bound to intersect somewhere on the coorinate plane. Therefore, if I am ever given a system of equations where each line has a different slope, I know there will be one solution.

Solve Systems by Graphing

 * One method of solving systems of equations is to carefully graph the equations on the same coordinate plane.

****Example 2a: Graph each system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it.**


 * y = -x + 8
 * y = 4x - 7


 * Remember, to graph a line, we must write the equation in slope-intercept form. We then start at the y-intercept and count our slope by using rise over run.

After graphing my system, I found the solution to be the point (3, 5). This is the solution because the point (3, 5) is the //only point// that lies on both lines. Therefore, if I were to plug in  3 for x and 5 for y into either equation, I will obtain true statement. This is a great method to use to check a solution.

The point of intersection of a system of equations is always the solution of the system.

****Example 2b: Graph each system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it.**


 * x + 2y = 5
 * 2x + 4y = 2

In order to solve a system by graphing, I must make sure both of my equations are in slope-intercept form. In this example, I need to solve both equations for **//y//** before I can graph.

Now, I can graph my system since both equations are in slope-intercept form.



The lines in this system are parallel. They will never intersect so there is **no solution**. There is not one ordered pair that will satisfy both equations.

**More Practice:**  **Helpful Links: [|Solve by Graphing - Purple Math] [|Solve by Graphing - Regents Prep]**