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2.
How to find
solutions for simultaneous equations
To solve simultaneous equations having two variables, first use a few
mathematical tricks to eliminate one variable. Then make the coefficient of
the other remaining variable to be 1. Sometimes,
while simplifying the equations, we get some new looking equations, these
again have to be solved logically as before. Keep in mind that the equality
in the equations has to maintained very carefully. Any operation done on one
side has to be done on the other side too.
Example 1:
x + y = 10, x - y = 2. Find
the values of x and y.
x + y =
10
x ñ y =
2
Add the LHS of the two equations together. Similarly
add the RHS also. This does not violate the equality.
x + y
+ x ñ y = 10 + 2
2x = 12
x = 6
This makes y = 4.
We can solve the same equation in another way, and
yet we will get the same results.
x + y =
10
x - y = 2
Subtract the LHS of the two equations and subtract
also the RHS of the two equations.
We get
x + y ñ (x-y)
= 10 ñ2
2y = 8
y = 4
Thus x = 6
The
solution of the equation is (x, y ) is (6, 4).
Example 2
: Solve the following simultaneous equation 3x - 7y = 10 and x + 2y = 5.
3x +
7y =
10
(equation 1)
x + 2y
= 5
(equation 2)
Step 1: Multiply eq (2) by 3 and subtract from eq
(1)
3x +
7y - (3x + 6y) = 10 - 15
7y - 6y =
- 5
Thus y = - 5
Step 2: Substitute the value of y = - 5 in one of
the equations, you will get x = 15 (both the equations will give the same
value of x, as these two equations are being solved simultaneously)
The same equation
can be solved in another way.
Step 1:
3x +
7y =
10
(equation 1)
x + 2y
= 5
(equation 2)
Multiply eq (1) by 2 and eq (2) by 7 and subtract.
We get
6x + 14y - ( 7x + 14y) = 20 ñ 35
-x = -15
x = 15
Step 2: Substitute this value of x
in any of the
equations, you get y = - 5.
Thus the solution to the given simultaneous equations
is (15, -5)
3.
Graphical
Method of solving simultaneous equations
Graphical method for solving simultaneous equations is very simple.
By putting various values of (x,y) in both the equations, plot on a graph
the lines representing the two equations. To draw a line representing the linear equation, first put x = 0 and
find y. This will give you one point on the x-axis. Then put y = 0 and find
x. This will give you one point on the y-axis. Draw a line through these two
points on a graph sheet. Do the same procedure for the second (linear)
equation. Where the lines intersect, the co-ordinate of the point is the
solution of the two given simultaneous equations.
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