Lesson 3–1
Example 1  Solve the System of Equations by Completing a Table
Solve the system of equations by completing a table.	
x - y = 1
2x – y = 4

Solve for y in each equation.
x - y = 1		y = x – 1 
2x – y = 4		y = 2x – 4

The solution of the system of equations is the ordered pair that satisfies both equations. 

Use a table to find the solution that satisfies both equations.
x	 = x – 1 		= 2x – 4 		(,)	(,)	
1	 = 1 – 1	0	= 2(1) – 4	-2	(1, 0)	(1, -2)	
2	 = 2 – 1	1	= 2(2) – 4	0	(2, 1)	(2, 0)	
3	 = 3 – 1	2	= 2(3) – 4	2	(3, 2)	(3, 2)	

The solution of the system of equations is (3, 2).

Example 2  Solve by Graphing
Solve the system of equations by graphing.	
x + y = 2
3x – y = 6

Write each equation in slope–intercept form.
x + y = 2		y = –x + 2
3x – y = 6		y = 3x – 6

The graphs appear to intersect at (2, 0).

Check	Substitute the coordinates into each equation.
x + y = 2	3x – y = 6	Original equations	
2 + 0  2	3(2) – 0  6	Replace x with 2 and y with 0.	
2 = 2  	6 – 0  6	Simplify.	
	6 = 6  		

The solution of the system is (2, 0).

Example 3  Break–Even Point Analysis
BUSINESS  Laura’s Copies has two options for buying photocopies. With Option 1, you can buy a card for $40 each year and then pay $0.03 per copy. With Option 2, you can just pay $0.05 per copy. For how many copies would the cost of Options 1 and 2 be the same? What is the cost?

Let x = the number of copies, and let y = the cost of the copies.

Cost of x copies 
using Option 1	    is	 cost per copy	plus	cost of card.	
y	 =	0.03x	+	40	

Cost of x copies using Option 2	   is	cost per copy	
y	=	0.05x	

The graphs intersect at (2000, 100). This is the break–even point. The cost of making copies is the same if you make 2000 copies. The cost is $100. 

If a person plans to make less than 2000 copies during the year, he or she should use Option 2. If a person plan to make more than 2000 copies during the year, he or she should use Option 1.

Example 4  Intersecting Lines
Graph the system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent.	
4y – x = 8
x + y = 6

Write each equation in slope-intercept form.
4y – x = 8		y = x + 2
x + y = 6		y = –x + 6

The graphs intersect at (4, 3). Since there is one solution, the system is consistent and independent.

Example 5  Same Line
Graph the system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent.	
x + y = 1
3x + 4y = 6

Write each equation in slope-intercept form.
x + y = 1		y = –x + 
3x + 4y = 6		y = –x + 

Since the equations are equivalent, their graphs are the same line. Any ordered pair representing a point on that line will satisfy both equations.
So, there are infinitely many solutions to this system. This system is consistent and dependent.

Example 6  Parallel Lines
Graph the system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent.	
3y = –7x + 6
6y + 14x = –24

3y = –7x + 6		y = –x + 2
6y + 14x = –24		y = –x – 4

The lines do not intersect. Their graphs are parallel lines. So, there are no solutions that satisfy both equations. This system is inconsistent.