Topic 3: Solving two simultaneous linear equations
Two simultaneous equations are two equations which contain 2 variables e.g. x and y and the two
equations are both satisfied by the same values of x and y.
There are two different methods to solve simultaneous equations.
(1)
Graphical method
The straight line corresponding to each equation is drawn. Where the straight lines intersect is
the solution of the two simultaneous equations.
Example 1
Use a graphical method to solve this pair of simultaneous equations;
5x + 2y = 20
Y = 2x + 1
Solution 1
5x + 2y = 20
When x = 0
5 x 0 + 2y = 20
2y = 20
y = 20 ÷ 2
y = 10
(0,10)
y = 2x + 1
When x = 0
y=2x0+1
y=1
(0,1)
When y = 0
5x + 2 x 0 = 20
5x = 20
x = 20 ÷ 5
x=4
(4,0)
When y = 0
0 = 2x + 1
-1 = 2x
-1 ÷ 2 = x
-0.5 = x
(-0.5,0)
Plot these two lines (see graph paper)
The solution lies where these two lines intersect
Solution is x = 2, y = 5
Simultaneous equations with no solution
Sometimes when the straight lines corresponding to the two simultaneous equations are drawn, the
lines are parallel so they do not intersect. This means that there is no solution.
Example 2
Use a graphical method to show that this pair of simultaneous equations do not have a solution;
y – 2x = 4
2y = 4x – 1
Solution 2
y – 2x = 4
When x = 0
y–2x0=4
y=4
(0,4)
2y = 4x – 1
When x = 0
2y = 4 x 0 – 1
2y = -1
y = -1 ÷ 2
y = -0.5
(0,-0.5)
When y = 0
0 – 2x = 4
-2x = 4
x = 4 ÷ -2
x = -2
(-2,0)
When y = 0
2 x 0 = 4x - 1
0 = 4x - 1
1 = 4x
1÷4=x
0.25 = x
(0.25,0)
See graph paper
(2)
Algebra method
We can also solve two simultaneous equations using the algebra method known as elimination.
Case 1: If the numbers in front of the x or y terms are the same size but different signs in both
equations, the equations are added to get rid of the x or y terms.
Example 1
Solve the simultaneous equations
2x – y = 5 ……………..(A)
3x + y = 10 …………….(B)
Solution 1
(A) + (B)
2x – y = 5
+ 3x + y = 10
5x
= 15
x = 15 ÷ 5
x=3
Substitute x = 3 into (A)
2x3–y=5
6–y=5
6–5=y
1=y
Solution is x = 3, y = 1
Case 2: If the numbers in front of the x and y terms are the same in both equations the equations
are added to get rid of the x or y terms.
Example 2
Solve the simultaneous equations
3x + y = 5 …………………(A)
3x – 2y = 8 …………………(B)
Solution 2
(A) – (B)
3x + y = 5
- 3x – 2y = 8
3y = -3
y = -3 ÷ 3
y = -1
Substitute y = -1 into (A)
3x – 1 = 5
3x = 5 + 1
3x = 6
x=6÷3
x=2
Solution is x = 2, y = -1
Case 3: If the number in front of the x or y terms are not the same size in both equations but
they can be made the same by multiplying one equation by a suitable number we must multiply to
make the x or y term the same size then add or subtract the equations as necessary.
Example 3
Solve these simultaneous equations
5x + 2y = 17 …………………….. (A)
3x – y = 8 …………………………(B)
Solution 3
2 x (B) gives
(A) + (C)
6x – 2y = 16 ……………………….(C)
5x + 2y = 17
6x – 2y = 16
11x
= 33
x = 33 ÷ 11
x=3
Substitute x = 3 into (A)
5 x 3 + 2y = 17
15 + 2y = 17
2y = 17 – 15
2y = 2
y=2÷2
y=1
Solution is x = 3, y = 1
Case 4: If the numbers in front of the x or y terms are not the same size in both the equations
they can be made the same size by multiplying both of the equations by suitable numbers then add
or subtract the numbers as necessary.
Example 4
Solve these simultaneous equations
2x + 3y = 2 ………………………..(A)
3x – 4y = 20 ………………………(B)
Solution 4
3 x (A) gives
2 x (B) gives
(C) – (D)
6x + 9y = 6 ………………………….(C)
6x – 8y = 40 ………………………..(D)
6x + 9y = 6
- 6x – 8y = 40
17y = -34
y = -34 ÷ 17
y = -2
Substitute y = -2 into (A)
2x + 3 x -2 = 2
2x – 6 = 2
2x = 2 + 6
2x = 8
x=8÷2
x=4
Solution is x = 4, y = -2