SYSTEMS OF LINEAR EQUATIONS IN TWO

VARIABLES

Let’s try this!For each pair of equations, Sketch the graph Describe the relationship between the

lines on the graph

13

yxyx

423

xyyx

33

yxyx

6223

yxyx

Systems of Linear Equations in Two Variables

System of Linear Equations is a set of two or more linear equations, which are to be treated simultaneously, generally to solve for values of the variables that satisfy all of the equations, if there are such values.

If a system has a solution, it is called consistent; if it has no solution, it is inconsistent.

If a system is made up of 2 equivalent equations (coincide), such system is called a dependent system; otherwise, it is independent.

In general, given the system of linear equations

If ,

the system is consistent and independent graph: 2 lines intersect at exactly one point Unique solution (pt. of intersection)

cbyaxCByAx

cC

bB

aA

In general, given the system of linear equations

If ,

the system is inconsistent Graph: Two distinct parallel lines. No solution

cbyaxCByAx

cC

bB

aA

In general, given the system of linear equations

If ,

the system is dependent graph: Two lines coincide and are actually the

same line infinitely many solutions; every solution of

either equation is a solution of the other

cbyaxCByAx

cC

bB

aA

Solving Systems of Equation:

1. Graphical Solution 2. Algebraic Solution

2.1 Elimination Method2.2 Substitution Method

3. Cramer’s Rule

Graphical Solution Use

Geometer’s SketchpadWzgrapherGraphing calculator

Calculator Key Strokes MODE Choose

5: EQN1:anX + bnY = cn

Input valuesEx. x + y = 3

x – y = 3○ a1 = 1 b1 = 1 c1 = 3○ a2 = 1 b2 = -1 c2 = 3

Press equal sign twiceX = 3, Y = 0

ELIMINATION METHOD Eliminate one variable by addition or

subtraction of the equations and then solve for the solution of the remaining variable.

Example: Find the solution to the given system of linear equations.

yx

yx1

3

Example: Solution: Write both equations in the same form (Ax + By = C).

Multiply one or both of the equations by appropriate numbers (if necessary) so that one of the variables will be eliminated by addition.

yx

yx1

3

13

yxyx

Example: Solution: Add the equations to get an equation in one variable.

Solve the equation in one variable.

yx

yx1

3

13

yxyx

402 yx

42 x 2 x

Example:

Solution: Substitute the value obtained for one variable into

one of the original equations to obtain the value of the other variable.

yx

yx1

3

3 yx

3)2( y

1y

yx 1y 1)2(

y1

Example:

Solution: Check the two values in both of the original

equations.

yx

yx1

3

3 yx

3)1()2(

33

yx 111)2(

11

Example: Find the solution to the given system of linear equations.

yx

yx32102

Example: Solution: Write both equations in the same form (Ax + By = C).

Multiply one or both of the equations by appropriate numbers (if necessary) so that one of the variables will be eliminated by addition.

yx

yx32102

23102

yxyx

233)102(

yxyx

233036

yxyx

Example: Solution: Add the equations to get an equation in one variable.

Solve the equation in one variable.

2807 yx

287 x 4 x

yx

yx32102

233036

yxyx

Example:

Solution: Substitute the value obtained for one variable into

one of the original equations to obtain the value of the other variable.

102 yx

10)4(2 y

2y

yx 32

y32)4(

y2

yx

yx32102

Example:

Solution: Substitute the value obtained for one variable into

one of the original equations to obtain the value of the other variable.

102 yx

10)4(2 y

2y

yx 32

y32)4(

y2

yx

yx32102

Example:

Solution: Check the two values in both of the original

equations.

102 yx

10)2()4(2

1010

yx 32

)2(32)4(

66

yx

yx32102

Exercises: Find the solution to the given systems of linear

equations.

54323

yxyx

1453332

yxyx

24212

yxyx

6293

yxyx

yxyx43723

2553

yxyx

SUBSTITUTION METHOD Solve one equation for one of the

variables and substitute this expression into the other equation. Then solve for the variable.

It is easiest to use the method of substitution when one of the coefficients in an equation is 1.

Example:Find the solution to the given system of linear equations.

Solution: Solve one of the equations for one variable in

terms of the other. Choose the equation that is easiest to solve for x or y.

13492

yxyx

92 yx

yx 29

Example:Find the solution to the given system of linear equations.

Solution: Substitute into the other equation to get an

equation in one variable.

13492

yxyx

134 yx

13)29(4 yy

Example:Find the solution to the given system of linear equations.

Solution: Solve for the remaining variable (if possible).

13492

yxyx

13)29(4 yy13836 yy

13836 yy

355 y 7 y

Example:Find the solution to the given system of linear equations.

Solution: Substitute the value just found into one of the

original equations to find the value of the other variable.

13492

yxyx

yx 29

)7(29 x

5 x

Example:

Solution: Check the two values in both of the original

equations.

92 yx

9)7(2)5(

99

134 yx

13492

yxyx

1)7(3)5(4

11

Exercises: Find the solution to the given systems of linear

equations.

yx

xy2212

4233

yxyx

45

yxyx

6323

yxyx

32xyxy

32

31

23

xy

yx