2. REVIEW OF
CARTESIAN COORDINATE
SYSTEM
Cartesian Coordinate System consists of:
two coplanar perpendicular number lines
y-axis or the
vertical line
x-axis or the
vertical line
. origin

3. REVIEW OF
CARTESIAN COORDINATE
SYSTEM
Cartesian Coordinate System consists of:
four regions called quadrants
Quadrant II Quadrant I
(–,+) (+,+)
.
Quadrant III Quadrant IV
(–,–) (+,–)

4. SYSTEMS OF LINEAR
EQUATIONS IN TWO
VARIABLES
A system of linear equations in two variables refers to
two or more linear equations involving two unknowns,
for which, values are sought that are common solutions
of the equations involved.
Example:
x–y=–1 (Eq. 1)
2x + y = 4 (Eq. 2)

5. SYSTEMS OF LINEAR
EQUATIONS IN TWO
VARIABLES
Just like in solving the linear equations, the system of linear
equations also have their solutions, wherein this time, the
solution is an ordered pair that makes both equations true.
To check whether the given ordered pair is the solution for the
system, simply substitute the values of x and y to the
equations then see whether both equations hold. (If the left
side of the equation is equal to its right side)

6. SYSTEMS OF LINEAR
EQUATIONS IN TWO
VARIABLES
From the previous example, check whether the ordered pair (1,2)
is the solution to the system.
For Eq. 1:
Remember:
x – y = – 1 ; (1,2) It is not enough to check
(1)– (2) = – 1 whether the given order
– 1 = –1  pair is true in one of the
given equations. You still
have to check the other
Eq. 1 is true in the equation to see if both
ordered pair (1,2) equations hold.

7. SYSTEMS OF LINEAR
EQUATIONS IN TWO
VARIABLES
For Eq. 2:
Since both equations hold,
2x + y = 4 ; (1,2) this implies that the point
2(1) +(2) = 4 (1,2) is a common point of
2 +2 =4 the lines whose equations
are x – y = – 1 & 2x + y = 4.
4=4 
Eq. 2 is also true in
the ordered pair (1,2) Hence, (1,2) is the point of
intersection of the lines.

8. SYSTEMS OF LINEAR
EQUATIONS IN TWO
VARIABLES
2x + y = 4
x–y=–1
(2,1)

9. Determine whether the given point is a solution of the
given system of linear equations.
a. (3,-1)
x–y=4 (Eq.1)
y = – 2x + 5 (Eq. 2)
For Eq. 1: For Eq. 2:
x – y =4 y = - 2x + 5
(3) – (-1) = 4 (-1) = - 2(3) + 5
3+1=4 -1 = -6 + 5
4=4  -1 = -1 
Since both of the equations hold, the solution of
the given system of linear equations is (3,-1).

10. y = -2x + 5
x–y=4
(3,-1)

11. Determine whether the given point is a solution of the
given system of linear equations.
b. (- 1,- 3)
2x – y = 1 (Eq.1)
2x + y = 5 (Eq. 2)
For Eq. 1: For Eq. 2:
2x – y = 1 ; (-1,-3) 2x + y = 5 ; (-1,-3)
2(-1) – (-3) = 1 2(-1) + (-3) = 5
-2 + 3 = 1 -2 – 3 = 5
1=1 -5≠-5
Since one of the equations doesn’t hold, the lines
of the equations will not meet @ point (-1,-3)

12. (-1,-3)

13. DIFFERENT
SYSTEMS OF
LINEAR
EQUATIONS

14. Geometrically, solutions of systems of linear equations are
the points of intersection of the graph of the equations.
INDEPENDENT
CONSISTENT
SYSTEMS OF DEPENDENT
LINEAR
EQUATIONS
INCONSISTENT

15. CONSISTENT - INDEPENDENT
SYSTEM
intersecting exactly one
lines (unique)
solution
a1 b1 c1
a2
≠ b ≠c
2 2

16. CONSISTENT - DEPENDENT
SYSTEM
coinciding infinitely
lines many
solutions
a1 = b1 = c1
a2 b2 c2

17. INCONSISTENT
SYSTEM
parallel
lines
no solution
a1 b1 c1
= ≠
a2 b2 c2

18. Without graphing, identify the kind of system, and state
whether the system of linear equations has exactly one
solution, no solution or infinitely many solutions.
a. x + 2y = 7 1 2 7 *consistent – independent
2x + y = 4 2
≠ 1≠ 4 *one unique solution
b. 4x = -y – 9 4 1 -9 *inconsistent
2y = -8x – 5 8
= 2≠ -5 *no solution
a. 3x + 4y = -12 3 4 -3 *consistent – dependent
y = - ¾x – 3 ¾
= 1= -3 *one unique solution

19. ASSIGNMENT:
• Look for the methods on how to solve the
solutions of the systems of linear equations.
END…