Lesson 5-1

Example 1 Number of Solutions	
	Use the graph at the right to determine whether each system has no solution, one solution, or infinitely many solutions.
	a. y = 2x + 2
	4x - 2y = 10
	Since the graphs are parallel, there are no solutions.

	b. y = 2x – 5
	4x - 2y = 10
	Since the graphs coincide, there are infinitely many solutions.

Example 2 Solve a System of Equations	
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.
a. y = x – 3
	x – 3y = -3
The graphs are parallel lines. Since they do not intersect, there are no solutions to this system of equations. Notice that the lines have the same slope but different y-intercepts. Recall that a system of equations that has no solution is said to be inconsistent.

	b. x – y = 3		
		2x – y = -1
The graphs appear to intersect at (-4, -7). Check by replacing x with –4 and y with –7 in each equation.
		Check	x – y = 3	2x – y = -1
			 -4 – (-7)  3	 2(-4) – (-7)  -1
	 -4 + 7  3	-8 + 7  -1
                             3 = 3  		-1 = -1	  
	The solution is (-4, -7).

	

Example 3 Write and Solve a System of Equations
CLUBS  Alexis bought pizza and soda for the ski club meeting. For one meeting she bought 4 pizzas and 10 sodas for $63. The next meeting she bought 3 pizzas and 8 sodas for $48. What is the cost of one pizza?

Let p = the cost of one pizza and let s = the cost of one soda. Write a system of equations to represent the situation.

total cost 

equals 

total cost of sodas

plus

Total cost of pizzas

Words      		     		   				     

Variables       Let p = number of pizzas.	Let s = number of sodas.
	
Equation	4p	+	10s	=	63
		3p	+	8s	=	48
					
Graph the equations 4p + 10s = 63 and 3p + 8s = 48. The graphs appear to intersect at the point with coordinates (12, 1.5). Check this estimate by replacing p with 12 and s with 1.5 in each equation.

CHECK:	  4p + 10s = 63	   3p + 8s = 48
	 4(12) + 10(1.5)  63	  3(12) + 8(1.5)  48
	48 + 15  63	   36 + 12  48
	  63 = 63  	     48 = 48  
One pizza will cost $12.