Download A Linear Equation System of Two Variables

Definition
Linear equation system of two variables can be written as follow
a1x + b1y = c1
a2x + b2y = c2
Where a1, a2, b1, b2, c1, and c2 are real numbers.
Determine the solution set of 2x + 3y = 6 and 2x + y = -2 by graphic
method!
Answer:
In equation 2x + 3y = 6
For
x=0→y=2
y=0→x=3
Thus, graphic 2x + 3y = 6 passes through the points (0, 2) and (3,
0).
In equation 2x + 2y = -2
For
x = 0 → y = -2
y = 0 → x = -1
Thus, graphic 2x + y = -2 passes through the points (0, -2) and (1,0).
From the graphic above, the two straight lines of the equations
intersect in one point, which is (-3,4). Then, the solution set is {(3,4)}.
To determine the solution of linear equation
system of two variables by elimination method, we
can use the following steps.
1)Choose the variable you want to eliminate first, and
then match the coefficients by multiplying the
equations by the appropriate number.
2)Eliminate one of the variables by adding or
subtracting the equations.
To be able to determine the solution of linear
equation system of two variables by elimination
method, study the following example.
Determine the solution set of the following linear equation
system
X + 3x = 1
2x – y = 9 by elimination method !
x + 3y = 1| x2 | = 2x + 6y = 2
2x – y = 9| x1 | = 2x – y = 9
7y = -7
Answer:
y = -1
x + 3y = 1 | x1 | = x + 3y = 1
2x – y = 9 | x3 | = 6x – 3y = 27
7x = 28
x =4
Thus, the solution set is {(4, -1)}
The idea of subtitution method is to solve one of the
equations for one of the variables, and plug this into the other
equation. The steps are as follows.
1)Solve one of the equations ( you choose which one) for one
of the variables (you choose which one).
2)Plug the equation in step 1 bavk into the other equation,
subtitute for the chosen variable and solve for the other. Then
you back-solve for the first variable.
To understand the solution of linear equation system of
two variables by subtitution method, study the following
example.
Determine the solution set of the following linear equation system
2x – 3y = -7
3x + 5y = -1 by subtitution method!
Answer:
2x – 3y = -7 → 3y = 2x + 7
y = 2x + 7
3
Then, subtitute y
= 2x + 7
into the second equation 3x + 5y = -1 to get the x value.
3
3x + 5 (2x + 7) = -1
3x + 10x + 35 = -1
3
↔ 9x + 10 x + 35 = -3
↔
3
19x = -38
9
↔
x = -2
Now we can subtitute this x value back into the expression
Y= 2(-2) + 7 = 1
3
Thus solution set is {(-2, 1)}
3
y
= 2x + 7, so we get
The mixed method is conducted by
eliminating one the variables (you choose
which one), and then subtituting the
result into one of the equations. This
method is regarded as the most effective
one is solving linear equation system. To
understand the method of finding the
solution, study the following example.
Determine the solution set of the following linear equation system
3x + 7x = -1
X – 3y = 5 by mixed method (elimination and subtitution)!
Answer:
3x + 5y = -1 | x1 | = 3x + 7y = -1
X – 3y = 5 | x3 | = 3x – 9y = 15
16y = -16
↔ y = -1
Then, subtitute y = -1 into the second equation x – 3y = 5 to get the x value.
x – 3(-1) = 5
↔x+3 =5
↔
x=2
Thus, the solution set is {(2, -1)}
 A general linear equation system of
three variables x,y, and z can be
written as.
 ax + by + cz = d where a, b, c, and d Є
R.
 Definition
 Linear equation system of three
variables can be written as follow.
 a1x + b1y + c1z = d1
 a2x + b2y + c3z = d3
 a3x + b3y + c3z = d3
 Where a1, a2, a3,b1, b2,, b3,c1, c2, c3, d1, d2,
and d3 are real numbers.
 Determine the solution set of the linear equation system below!
 2x – 3y + 2z = -1
 3x + 2y – z = 10
 -4x – y – 3z = -3




Answer:
2x – 3y + 2z = -1
3x + 2y – z = 10
-4x – y – 3z = -3
… (1)
… (2)
… (3)
 From equations (1) and (3), eliminate x, so we have
 2x – 3y + 2z = -1
 3x + 2y – z = 10
| x2 | 4x – 2y +4z = -2
| x1 | -4x – y – 3z = -3
-3y + z
= -5
… (4)
Y= x2 + 3x - 18
Y = -x2 + x + 6