Graphing Systems of Linear Equations
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1 Graphing Systems of Linear Equations Groups of equations, called systems, serve as a model for a wide variety of applications in science and business. In these notes, we will be concerned only with groups of equations that are linear, that is, equations that may be written in the general form Ax + By = C. The following are examples of systems of linear equations in two variables. Notice that the equations are linear although they may not be written in general form. y = 7 5x 8y 7 y x + 9 5y = 4 y = 2( x 5) y 1 2( y + 4) = 8 Solution to a System of Equations: A solution of a system of two linear equations in two variables is an ordered pair that satisfies both equations, that is, it makes both equations in the system TRUE. Example: Is the ordered pair (4, -1) a solution to the system x 2y y? Solution: To determine if the ordered pair is a solution we must substitute the values of the x-coordinate and y- coordinate into both equations and determine if the resulting equations are true. x 2y 4 2( 1) 4 6 y 2(4) ( 1) Because both equations are true, the ordered pair is a solution to the system of linear equations. Example: Is the ordered pair (1, -1) a solution to the system + y + 4y? Solution: To determine if the ordered pair is a solution we must substitute the values of the x-coordinate and y- coordinate into both equations and determine if the resulting equations are true. + y 3(1) y 2(1) + 4( 1) Because the ordered pair only satisfies the first equation but not the second equation, it is not a solution to the system.
2 Types of Solutions: A system of linear equations in two variables may have any one of three different solution types: one solution, no solution, or an infinite number of solutions. To justify these solution types, it will help to look at the graph of a system. Consistent & Independent System: x + y = 5 + y = 3 slope-intercept By examining the graph, it is clear that the equations intersect at a single point. This point is an ordered pair that lies on both lines. Consequently, the point of intersection is the ordered pair that is a solution to both equations and is therefore a solution to the system. A system with at least one solution is called a consistent system. A consistent system with exactly one solution is an independent system. Inconsistent System: + y = 5 y = 4 slope-intercept By examining the graph, it is clear that the equations will never intersect because the lines are parallel. Consequently, because there is no ordered pair that satisfies both equations, there is no solution to the system. A system with no solutions is called an inconsistent system.
3 Consistent & Dependent System: 4x y = 10 y = slope-intercept Although it appears as though there is only one line, there are actually two. Both lines lie directly on top of the other. Because every ordered pair on one line also lies on the other, there are an infinite number of solutions. A system with at least one solution is called a consistent system. A consistent system with exactly one solution is a dependent system. A quick way to determine the type of solutions is to compare the slope and y-intercept of each line. 1. If the slopes are different the system has one solution and is independent. 2. If the slopes are the same, then compare the y-intercepts: a. If the y-intercepts are different, then there is no solution (inconsistent system.) b. If the y-intercepts are the same, then there are an infinite number of solutions (dependent system). Example: By examining the slope and y-intercept, determine the type of solutions to the system. a. 3 y = 4x b. y = 5x 3 y = 5x c. y = 5x y = 5x Solution: a. Slopes are different, exactly one solution (independent system). b. Slopes are the same, y-intercepts are different, so no solution (inconsistent system.) c. Slopes are the same; y-intercepts are the same, so infinite solutions (dependent system.) Example: Determine if the system 2y = 5 y = 3 2 x + 3 is independent, dependent or inconsistent. Solution: Write the first equation id slope intercept form to obtain y x 5. Because the slopes are the 2 2 same but the y-intercepts are different, the system is inconsistent. There are no solutions.
4 Solving a System by Graphing: When solving a system by graphing, we are looking for the point of intersection of the two graphs (if there is one.) x + y x y = 4 Solution: Graph both equations on the same coordinate plane using any acceptable method (table, intercepts, slopeintercept The easiest method of graphing these lines is to use the slope-intercept method. Solving both equations for y we have: y = x y = x 4 The lines intersect at the point (3, -1) x 3y = 15 Solution: Graph both equations on the same coordinate plane using any acceptable method (table, intercepts, slopeintercept The easiest method of graphing these lines is to use the slopeintercept method. Solving both equations for y we have: Since both equations have the same slope we know that the lines are parallel. There is no solution.
5 6x = 15 3y Solution: Graph both equations on the same coordinate plane using any acceptable method (table, intercepts, slopeintercept The easiest method of graphing these lines is to use the slope-intercept method. Solving both equations for y we have: Since both equations have the same slope and the same y- intercept, they are essentially the same line and therefore there are an infinite number of solutions. Applications: Example: A demand curve for a particular product expresses the relationship between the price of that product and the quantity demanded and is given by the equation Q = 17 2 p. The supply curve expresses the relationship between price and the quantity that the company will make available for sale and is given by the equation Q p + 1. Graph the supply and demand curves and find the equilibrium point. Solution: The equilibrium point is where the supply curve intersects the demand curve. This point represents the price and quantity that a product should be sold at in order to maximize sales. The equilibrium point is (4, 9)
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