1. Algebra 2 3.1 Using Graphs & Tables to Solve Linear Systems
A system of equations is a set of two or more equations containing two or more variables. A linear system is a system of equations containing only linear equations.
Recall that a line is an infinite set of points that are solutions to a linear equation. The solution of a system of equations is the set of all points that satisfy each equation.
Algebra Using Graphs & Tables to Solve Linear Systems
On the graph of the system of two equations, the solution is the set of points where the lines intersect. A point is a solution to a system of equation if the x- and y-values of the point satisfy both equations.
Ex 1:
Use substitution to determine if the given ordered pair is an element of the solution set for the system of equations.
(1, 3);
x – 3y = –8
3x + 2y = 9
Because the point is a solution for both equations, it is a solution of the system.
3x + 2y = 9
3(1) +2(3) 9 
x – 3y = –8
(1) –3(3) –8 
Substitute 1 for x and 3
for y in each equation.

2. Algebra 2 3.1 Using Graphs & Tables to Solve Linear Systems
Ex 2:
Use a graph and a table to solve the system. Check your answer.
2x – 3y = 3
y + 2 = x
y= x – 2
y= x – 1
Solve each equation for y.
On the graph, the lines appear to intersect at the ordered pair (3, 1)

3. Algebra 2 3.1 Using Graphs & Tables to Solve Linear Systems
y= x – 1
y= x – 2
Make a table of values for each equation. Notice that when x = 3, the y-value for both equations is 1. 1 3 2 –1 y x x y –2 1
– 1 2 3
The solution to the system is (3, 1).

4. Algebra 2 3.1 Using Graphs & Tables to Solve Linear Systems
The systems of equations in Example 2 have exactly one solution. However, linear systems may also have infinitely many or no solutions. A consistent system is a set of equations or inequalities that has at least one solution, and an inconsistent system will have no solutions.
You can classify linear systems by comparing the slopes and y-intercepts of the equations. An independent system has equations with different slopes. A dependent system has equations with equal slopes and equal y-intercepts.

5. Algebra 2 3.1 Using Graphs & Tables to Solve Linear Systems
Ex 3:
Classify the system and determine the number of solutions.
x = 2y + 6
3x – 6y = 18
y = x – 3
The equations have the same slope and
y-intercept and are graphed as the same line.
Solve each equation for y.
The system is consistent and dependent with infinitely many solutions.

6. Algebra 2 3.1 Using Graphs & Tables to Solve Linear Systems
Ex 4:
City Park Golf Course charges $20 to rent golf clubs plus $55 per hour for golf cart rental. Sea Vista Golf Course charges $35 to rent clubs plus $45 per hour to rent a cart. For what number of hours is the cost of renting clubs and a cart the same for each course?
Step 1 Write an equation for the cost of renting clubs and a cart at each golf course.
City Park Golf Course: y = 55x + 20
Sea Vista Golf Course: y = 45x + 35
Because the slopes are different, the system is independent and has exactly one solution.

7. Algebra 2 3.1 Using Graphs & Tables to Solve Linear Systems
Step 2 Solve the system by using a table of values.
Use increments of to represent 30 min.
y = 55x + 20
y = 45x + 35
When x = , the y-values are both The cost of renting clubs and renting a cart for hours is $ at either company. So the cost is the same at each golf course for hours. x y 20
47.5 1 75
102.5 2
120 x y 35
57.5 1 80
102.5 2
125