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SYSTEMS OF LINEAR
EQUATIONS IN TWO
VARIABLES
REVIEW OFCARTESIAN
COORDINATE SYSTEMCartesian Coordinate System consists of:
y-axis or the vertical line
x-axis or the vertical line
two coplanar perpendicular number lines
. origin
Cartesian Coordinate System consists of:
.
four regions called quadrants
Quadrant I
(+,+)
Quadrant II
(–,+)
Quadrant III
(–,–)
Quadrant IV
(+,–)
REVIEW OFCARTESIAN
COORDINATE SYSTEM
SYSTEMS OF LINEAR EQUATIONS IN TWO
VARIABLESA 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)
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)
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:x – y = – 1 ; (1,2)(1)– (2) = – 1 – 1 = –1
Eq. 1 is true in the ordered pair (1,2)
Remember:It is not enough to check whether the given order pair is true in one of the given equations. You still have to check the other equation to see if both
equations hold.
SYSTEMS OF LINEAR EQUATIONS IN TWO
VARIABLES
For Eq. 2:2x + y = 4 ; (1,2)
2 + 2 = 42(1) +(2) = 4
4 = 4
Eq. 2 is also true in the ordered pair (1,2)
Since both equations hold, this implies that the point (1,2) is a common point of the lines whose equations are x – y = – 1 & 2x + y = 4.
Hence, (1,2) is the point of intersection of the lines.
SYSTEMS OF LINEAR EQUATIONS IN TWO
VARIABLES
2x + y = 4x – y = – 1
(2,1)
SYSTEMS OF LINEAR EQUATIONS IN TWO
VARIABLES
a. (3,-1) x – y = 4 (Eq.1) y = – 2x + 5 (Eq. 2)
For Eq. 1:x – y = 4 (3) – (-1) = 4 3 + 1 = 4 4 = 4
Determine whether the given point is a solution of the given system of linear equations.
For Eq. 2: y = - 2x + 5 (-1) = - 2(3) + 5 -1 = -6 + 5 -1 = -1
Since both of the equations hold, the solution of the given system of linear equations is (3,-1).
x – y = 4
y = -2x + 5
(3,-1)
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. 1:2x – y = 1 ; (-1,-3)2(-1) – (-3) = 1 -2 + 3 = 1 1 = 1
For Eq. 2:2x + y = 5 ; (-1,-3)2(-1) + (-3) = 5 -2 – 3 = 5 - 5 ≠ - 5
Since one of the equations doesn’t hold, the lines of the equations will not meet @ point (-1,-3)
(-1,-3)
DIFFERENT SYSTEMS OF
LINEAR EQUATIONS
Geometrically, solutions of systems of linear equations are the points of intersection of the graph of the equations.
SYSTEMS OF LINEAR
EQUATIONS
CONSISTENT
INCONSISTENT
INDEPENDENT
DEPENDENT
CONSISTENT - INDEPENDENT SYSTEM
intersecting lines
exactly one (unique) solution
a1 b1 c1
a2 ≠
b2 ≠
c2
CONSISTENT - DEPENDENT SYSTEM
coinciding lines
infinitely many
solutions
a1 = b1 = c1
a2 b2 c2
INCONSISTENT SYSTEM
parallel lines no solution
a1 b1 c1
a2 =
b2 ≠
c2
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 2x + y = 4 1 2 7
2 ≠
1 ≠
4
*consistent – independent *one unique solution
b. 4x = -y – 9 2y = -8x – 5 4 1 -9
8 =
2 ≠
-5
*inconsistent*no solution
a. 3x + 4y = -12 y = - ¾x – 3 3 4 -3
¾ =
1 =
-3
*consistent – dependent *one unique solution
ASSIGNMENT:
• Look for the methods on how to solve the solutions of the systems of linear equations.
END…
I HOPE YOU LEARNED
THANK YOU &GOD BLESS US!