Two linear equations in two variables are said to be simultaneous if they are considered at the same time.
Solution of Linear Simultaneous Equations
Simultaneous equations are solved exactly either by the substitution method or the elimination method. An approximate solution can be found by using the graphical method.
Substitution Method
To solve simultaneous equations, find the value of y in terms of x (or vice versa) for one of the two equations and then substitute this value into the other equation.
Example 1
Solve the following simultaneous equations by using the substitution method:
Solution:
Label the equations as follows:
From (1) we have:
y = 3x ...(3)
Substituting y = 3x in (2) gives:
So, the solution is (2, 6).
The Graphical Method
The graphical solution of the simultaneous equations
is given by the point of intersection of the graphs.
Consider the graph of
It passes through the origin (0, 0) and the point (1, 3).
Consider the graph of x + y = 8.
x-intercept: When y = 0, x = 8.
y-intercept: When x = 0, y = 8.
The lines intersect at (2, 6). So, the solution is (2, 6) as shown in the diagram.
To solve the simultaneous equations, make the coefficients of one of the variables the same value in both equations. Then either add the equations or subtract one equation from the other (whichever is appropriate) to form a new equation that contains only one variable. This is referred to as eliminating the variable.Solve the equation thus obtained. Then substitute the value found for the variable in one of the given equations and solve it for the other variable. Write the solution as an ordered pair.
Example 2
Solve the following simultaneous equations by using the elimination method:
Solution:
Label the equations as follows:
Notice that 3y appears in the left-hand side of both equations. Adding the left-hand side of (1) and (2), and then the right-hand sides, gives:
Note:
We have added equals to equals, and addition eliminates y.
Substituting x = 3 in (1) gives:
So, the solution is (3, 3).
Example 3
Solve the following simultaneous equations by using the elimination method:
Solution:
Label the equations as follows:
Subtracting (2) from (1) gives:
Substituting y = 4 in (1) gives:
So, the solution is (5, 4).
Example 4
Solve the following simultaneous equations by using the elimination method:
Solution:
Label the equations as follows:
Multiplying (1) by 2 and (2) by 3 gives:
Subtracting (3) from (4) gives:
Substituting x = 2 in (1) gives:
So, the solution is (2, 3).
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