4. EXAMPLE
Determine whether the given point is a solution of the following system.
point: (– 3, 1)
system: x – y = – 4
2x + 10y = 4
Plug the values into the equations.
First equation: – 3 – 1 = – true
Second equation: 2(– 3) + 10(1) = – = true
Since the point (– 3, 1) produces a true statement in both equations, it is a solution.

5. EXAMPLE
Determine whether the given point is a solution of the following system
point: (4, 2)
system: 2x – 5y = – 2
3x + 4y = 4
Plug the values into the equations
First equation: 2(4) – 5(2) = 8 – 10 = – true
Second equation: 3(4) + 4(2) = = 20  false
Since the point (4, 2) produces a true statement in only one equation, it is NOT a solution.

6. (3, 5) Graph Type of System Number of Solutions
If the lines intersect, the system of equations has one solution given by the point of intersection.
Consistent System
The equations are independent.
(3, 5)
Two lines intersect at one point.
If the lines are parallel, then the system of equations has no solution because the lines never intersect.
Inconsistent System
The equations are independent.
Parallel lines
If the lines lie on top of each other, then the system has infinitely many solutions. The solution set is the set of all points on the line.
Consistent System
The equations are dependent.
Lines coincide

12. Example
Without graphing, determine the number of solutions of the system. 3x + y = 1
3x + 2y = 6
Write each equation in slope-intercept form.
3x + y = First equation
y = –3x Subtract 3x from both sides.
3x + 2y = Second equation
2y = –3x Subtract 3x from both sides.
Divide both sides by 2.
The lines are intersecting lines (since they have different slopes), so the system is consistent and independent and has one solution.

13. Example
Without graphing, determine the number of solutions of the system x + y = 0
2y = –6x
Write each equation in slope-intercept form.
3x + y = 0 First equation
y = –3x Subtract 3x from both sides.
2y = –6x Second equation
y = –3x Divide both sides by 2.
The two lines are identical, so the system is consistent and dependent, and there are infinitely many solutions.