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!