Equations
Any expression F(X) when equated to an expression or a constant is called an equation.
When the highest power of X is one it is called a simultaneous equation.
What are Simultaneous Equations
A set of equations having many variables, is called simultaneous equations. To solve a simultaneous equation of two variables we need minimum two equations, which are different from each other.
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| Simultaneous Equation |
Solving Simultaneous Equations:
Substitution method
Algorithm:
Step 1: Express one of the variables in terms of the other.
Step 2: Substitute the value of the variable in the second equation to obtain an equation in terms of one single variable.
Step 3: Solve for the variable.
Step 4: Substitute the value of the variable in the expression to get the solution of the other variable.
Solved Example:
Solve 2x +3y =5; 5x +2y=7
Step 1: 2x= 5-3y; x= (5-3y)/2
Step 2: Substituting the value of x in the second equation we get
5((5-3y)/2) + 2y=7 Solving for y we get
(25-15y+4y)/2=7
(25-11y)/2=7
25-11y=14
25-14=11y
11=11y
y=1
Step 3: Substituting the value of x in step 1 we get x=(5-3(1))/2 =1
Hence the solution set is X=1, y=1 or {1, 1}
Method 2:
Elimination Method:
Algorithm:
Step 1: Make the coefficients of one of the variables same.
Step 2: If the signs are same, subtract and if the signs are different add them to obtain a linear equation of one variable and solve for it.
Step 3: Substitute the obtained value in any one of the equations to solve for the other variable.
Solved Example:
4x+ 3y = 11; 3x +2y=8
4x+3y=11 --------> multiply 2
3x+2y=8 --------> multiply 3
8x + 6y =22
9x + 6y =24 (as the signs are same subt)
(-) (-) (-)
-x = -2 or x=2
Substituting the value of x in the first equation we get
4(2) +3y =11;
8+3y=11;
3y= 11-8;
3y=3; y=1.
The solution set is x= 2 and y = 1.
Cross Multiplication Method
Algorithm:
Step 1: Rearrange the equations as shown.
If the equations are as follows
Step 2: Cross-multiply and subtract starting from top end corner of the denominator and write as follows
Step 3:
Equate the x term with constant term and solve for x. Similarly equate y term with constant term and solve for y.
Solved Example:
2x +3y = 7
6x -5y =11
Cross-multiplying and subtracting we get
