SYSTEMS OF LINEAR EQUATIONS
Systems of linear equations refer to a set of two or more linear equations used to find the value of
the unknown variables. If the set of linear equations consist of two equations there will be two
unknown variables. If the set consists of three equations there will be three variables and so on. If
there are more variables present than linear equations, the system cannot be solved.
There are three general methods for solving systems of linear equations: graphically, by method of
elimination and by substitution. Remember that in order to be able to solve the system, it is
necessary that you have the same number of equations and unknown variables.
Solving a System of Linear Equations Graphically
If you are given a system of two linear equations with two unknowns the system can be solved and
will have two answers, one for each of the variables. The graph of each linear equation will be a
straight line, and the point of intersection of the two straight lines represents the solution to the
system of equations. Thus, the solution to a system of two linear equations with two unknowns is an
ordered pair of numbers ( x, y ) . It is called a consistent system.
Example:
2x + 3y =
5
5x + 2 y =
7
Thus, the solution is x = 1 and y = 1, or (1, 1).
Method of Elimination
This method consists on eliminating one variable by addition or subtraction of the linear equations.
To be able to cancel a variable, the variable needs to have the same coefficient in both equations,
however, most of the time this will not be the case. If the variables have a different coefficient,
multiply the coefficients of each equation with the opposite equation.
Example:
3x − y = 4
− 4x + 2 y = 2
Step 1 – Start by canceling one of the variables. You can choose whichever
variable you want to start with. In this case we are going to start by
cancelling the x variable.
To cancel the x multiply the coefficient of x in the first equation with
the entire second equation, and vice versa. For this system we do not
need to worry about changing signs since they are different already.
4(3 x − y = 4 )
3(− 4 x + 2 y = 2 )
→
12 x − 4 y = 16
− 12 x + 6 y = 6
Step 2 – Add both equations together to cancel the x and solve for y.
12 x − 4 y = 16
− 12 x + 6 y = 6
2 y = 22
22
y=
2
y = 11
Step 3 – Use the variable that we just found, y, and plug it into any of the
equations to obtain x.
3x − y = 4
3 x − 11 = 4
3 x = 4 + 11
15
3
x=5
x=
The solution is x = 5 and y = 11, or (5, 11).
Method of Substitution
The method of substitution consists of solving one of the equations for any variable and then
substituting the resultant equation into the other equation, thus leaving a one-equation, oneunknown system.
Example:
3x − y = 4
− 4x + 2 y = 2
Step 1 – Pick one equation and solve for one variable. For this system we are
going to use the first equation and solve for y.
3x − y = 4
3x − 3x − y = 4 − 3x
− y = 4 − 3x
y = 3x − 4
Step 2 – Use the new equation to substitute the value of y into the second
equation.
− 4x + 2 y = 2
− 4 x + 2(3 x − 4 ) = 2
− 4x + 6x − 8 = 2
2x − 8 + 8 = 2 + 8
2 x = 10
10
2
x=5
x=
Step 3 – Plug in the value of the x variable into the equation obtained in step
1.
y = 3x − 4
y = 3(5) − 4
y = 15 − 4
y = 11
The solution is x = 5 and y = 11, or (5, 11).
Parallel Lines
If two lines are parallel they have the same slope and thus no point of intersection since they run in
the same direction and they never meet. When a system of linear equations consist of two parallel
lines the system is said to have no solution since the lines have no point of intersection. The system
is said to be inconsistent.
Example:
2 x + 3 y =1 → − 2 ( 2 x + 3 y =1) → − 4 x − 6 y =−2
4x + 6 y = 7 →
4x + 6 y = 7 →
← Parallel lines
4x + 6 y = 7
0 + 0 =5
← No solution
Coinciding Lines
When a system of linear equations consist of two lines that have the same straight line when
graphed, then the two equations are equivalent and the lines are said to be coinciding lines. Since
they are basically the same line, any point that satisfies one equation will satisfy the second
equation. Therefore the system has an infinite number of solutions. The system is said to be
dependent.
Example:
2x − 4 y = 8
→
2x − 4 y = 8
→
x − 2y =
4 → − 2( x − 2y =
4) →
2x − 4 y =
8
−2 x + 4 y =
−8
0=0
← Coinciding lines
← Infinite number of
solutions
SYSTEM OF LINEAR EQUATIONS – EXERCISES
1.
3.
3x + 4 y =
10
−6 x + 3 y =
−9
2x + 4 y =
−12
3x + 5 y =
−16
7x − 2 y =
2
5. 1
1
1
x− y =
2
7
7
2.
4.
7x + 4 y =
−5
−2 x + 5 y =
26
x + 2y =
5
2x + 4 y =
1
6.
6x + 5 y =
28
7x + 2 y =
2
3x + 2 y =
1
4x − 3y =
2
7.
3x − 2 y =
1
9x − 6 y =
3
8.
9.
3x − 2 y =
4
12 x − 8 y =
6
10.
2x + 4 y =
11
6x + 2 y =
3
SYSTEM OF LINEAR EQUATIONS – ANSWERS TO EXERCISES
1.
3 x + 4 y = 10 → 6 x + 8 y = 20
−6 x + 3 y =
−9 → −6 x + 3 y =
−9
2.
7x + 4 y =
−5 → 14 x + 8 y =
−10
−2 x + 5=
y 26 → −14 x + 35=
y 182
43 y = 172
y=4
11 y = 11
y =1
3 x + 4 (1) = 10 → x = 2
Solution (2, 1)
3.
2x + 4 y =
−12
3 x + 5 y =−16
7 x + 4 ( 4) =
−5 → x =
−3
Solution (–3, 4)
→
6 x + 12 y =
−36
→ −6 x − 10 y =32
4.
x + 2y =
5
2x + 4 y = 2
→ − 2x − 4 y =
−10
6→ 2 x + 4 y = 1
2y = −4
y = −2
0=9
2 x + 4 ( −2 ) =
−12 → x =
−2
Solution (–2, –2)
5.
56
7x − 2 y = 2
→
7x – 2y = 2
1
1
1
x− y =
→ –7x + 2y = –2
2
7
7
No Solution
6.
6x + 5y = 28 →
7x + 2y = 2
12x + 10y =
→ –35x – 10y = –
10
0+0=0
Lines Coincide
Infinitely Many Solutions
–23x = 46
x = –2
6(–2) + 5y = 28 → y =
8
Solution (–2, 8)
7. 3x – 2y = 1
9x – 6y = 3
x=7
→
→
–9x + 6y = –3
9x – 6y = 3
0+0=0
Lines Coincide
8.
17
3x + 2y = 1
4x – 3y = 2
→
→
1
( 17 ) + 2 y =→
3 7
Infinitely Many Solutions
9x + 6y = 3
8x – 6y = 4
17x = 7
y=
−2
(
Solution 7
9.
3x – 2y = 4 →
12x – 8y = 6 →
–12x + 8y = – 16
12x – 8y = 6
0 = –10
No Solution
10.
17
17
, −2
17
)
2x + 4y = 11 → 2x + 4y = 11
6x + 2y = 3 → –12x – 4y = –6
–10x = 5
x= −1
2
2(− 1 2 ) + 4 y = 11 → y = 3
Solution − 1 , 3
2
(
)