05 systems of equations in two variables
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Transcript of 05 systems of equations in two variables
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