Simultaneous Equations
Simultaneous equations are two or more algebraic equations that share common variables and are solved at the same time (that is, simultaneously). For example, equations x + y = 5 and x  y = 6 are simultaneous equations as they have the same unknown variables x and y and are solved simultaneously to determine the value of the variables. We can solve simultaneous equations using different methods such as substitution method, elimination method, and graphically.
In this article, we will explore the concept of simultaneous equations and learn how to solve them using different methods of solving. We shall discuss the simultaneous equations rules and also solve a few examples based on the concept for a better understanding.
What are Simultaneous Equations?
Simultaneous equations are two or more algebraic equations with the same unknown variables and the same value of the variables satisfies all such equations. This implies that the simultaneous equations have a common solution. Some of the examples of simultaneous equations are:
 2x  4y = 4, 5x + 8y = 3
 2a  3b + c = 9, a + b + c = 2, a  b  c = 9
 3x  y = 5, x  y = 4
 a^{2} + b^{2} = 9, a^{2}  b^{2} = 16
We can solve such a set of equations using different methods. Let us discuss different methods to solve simultaneous equations in the next section.
Solving Simultaneous Equations
We use different methods to solve simultaneous equations. Some of the common methods are:
Simultaneous equations can have no solution, an infinite number of solutions, or unique solutions depending upon the coefficients of the variables. We can also use the method of cross multiplication and determinant method to solve linear simultaneous equations in two variables. We can add/subtract the equations depending upon the sign of the coefficients of the variables to solve them.
To solve simultaneous equations, we need the same number of equations as the number of unknown variables involved. We shall discuss each of these methods in detail in the upcoming sections with examples to understand their applications properly.
Simultaneous Equations Rules
To solve simultaneous equations, we follow certain rules first to simplify the equations. Some of the important rules are:
 Simplify each side of the equation first by removing the parentheses, if any.
 Combine the like terms.
 Isolate the variable terms on one side of the equation.
 Then, use the appropriate method to solve for the variable.
Solving Simultaneous Equations Using Substitution Method
Now that we have discussed different methods to solve simultaneous equations. Let us solve a few examples using the substitution method to understand it better. Consider a system of equations x + y = 4 and 2x  3y = 9. Now, we will find the value of one variable in terms of another variable using one of the equations and substitute it into the other equation. We have
x + y = 4  (1)
2x  3y = 9  (2)
From (1), we have
x = 4  y  (3)
Substituting this in (2), we get
2(4  y)  3y = 9
⇒ 8  2y  3y = 9
⇒ 8  5y = 9
Isolating the variable term to one side of the equation, we have
⇒ 5y = 9  8
⇒ y = 1/(5)
= 1/5
Substituting the value of y in (3), we have
x = 4  (1/5)
= 4 + 1/5
= (20 + 1)/5
= 21/5
Answer: So, the solution of the simultaneous equations x + y = 4 and 2x  3y = 9 is x = 21/5 and y = 1/5.
Solving Simultaneous Equations By Elimination Method
To solve simultaneous equations by the elimination method, we eliminate a variable from one equation using another to find the value of the other variable. Let us solve an example to understand find the solution of simultaneous equations using the elimination method. Consider equations 2x  5y = 3 and 3x  2y = 5. We have
2x  5y = 3  (1)
and 3x  2y = 5  (2)
Here, we will eliminate the variable y, so we find the LCM of the coefficients of y. LCM (5, 2) = 10. So, multiply equation (1) by 2 and equation (2) by 5. So, we have
[ 2x  5y = 3 ] × 2
⇒ 4x  10y = 6  (3)
[ 3x  2y = 5 ] × 5
⇒ 15x  10y = 25  (4)
Now, subtracting equation (3) from (4), we have
(15x  10y)  (4x  10y) = 25  6
⇒ 15x  10y  4x + 10y = 19
⇒ (15x  4x) + (10y + 10y) = 19
⇒ 11x + 0 = 19
⇒ x = 19/11
Now, substituting this value of x in (1), we have
2(19/11)  5y = 3
⇒ 38/11  5y = 3
⇒ 5y = 38/11  3
⇒ 5y = (38  33) / 11
⇒ y = 5/(11×5)
= 1/11
So, the solution of the simultaneous equations 2x  5y = 3 and 3x  2y = 5 using the elimination method is x = 19/11 and y = 1/11.
Solving Simultaneous Equations Graphically
In this section, we will learn to solve the simultaneous equations using the graphical method. We will plot the lines on the coordinate plane and then find the point of intersection of the lines to find the solution. Consider simultaneous equations x + y = 10 and x  y = 4. Now, find two points (x, y) satisfying for each equation such that the equation holds.
For x + y = 10, we have
x + y = 10  

x  0  10 
y  10  0 
So, we have coordinates (0, 10) and (10, 0). Plot them and join the points and plot the line x + y = 10.
For equation x  y = 4, we have
x  y = 4  

x  0  4 
y  4  0 
So, we have coordinates (0, 4) and (4, 0). Plot them and join the points and plot the line x  y = 4.
Now, as we have plotted the two lines, find their intersecting point. The two lines x + y = 10 and x  y = 4 intersect each other at (7, 3). So, we have found the solution of the simultaneous equations x + y = 10 and x  y = 4 graphically which is x = 7 and y = 3.
Important Notes on Simultaneous Equations
 Simultaneous equations are two or more algebraic equations that share common variables and are solved at the same time.
 Simultaneous equations can be solved using different methods such as substitution method, elimination method, and graphically.
 We can also use the cross multiplication and determinant method to solve simultaneous linear equations in two variables.
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Simultaneous Equations Examples

Example 1: Solve the simultaneous equations 2x  y = 5 and y  4x = 1 using the appropriate method.
Solution: To solve 2x  y = 5 and y  4x = 1, we will use the elimination method as it is easy to eliminate the variable y by adding the two equations. So, we have
2x  y = 5  (1)
y  4x = 1  (2)
Adding (1) and (2), we get
(2x  y) + (y  4x) = 5 + 1
⇒ 2x  y + y  4x = 6
⇒ 2x = 6
⇒ x = 6/2
= 3
Substitute this value of x in (1)
2(3)  y = 5
⇒ 6  y = 5
⇒ y = 6  5
= 11
Answer: Solution of simultaneous equations 2x  y = 5 and y  4x = 1 is x = 3 and y = 11.

Example 2: Find the solution of the simultaneous equations 2x  4y + z = 2, x + 5y  3z = 7, 3x + 2y  z = 10 using the substitution method.
Solution: We have
2x  4y + z = 2  (1)
x + 5y  3z = 7  (2)
3x + 2y  z = 10  (3)
From (1), we have
z = 2  2x + 4y
Substituting this value of z in (2) and (3),
x + 5y  3(2  2x + 4y) = 7
⇒ x + 5y  6 + 6x  12y = 7
⇒ 7x  7y = 13  (4)
3x + 2y  (2  2x + 4y) = 10
3x + 2y  2 + 2x  4y = 10
⇒ 5x  2y = 12  (5)
Now, solving the twovariable equations (4) and (5), multiply (4) by 2 and (5) by 7, we have
[7x  7y = 13 ] × 2 and [5x  2y = 12 ] × 7
⇒ 14x  14y = 26 and 35x  14y = 84
Now, subtracting the above two equations, we have
(14x  14y)  (35x  14y)= 26  84
⇒ 14x  35x  14y + 14y = 58
⇒ 21x = 58
⇒ x = 58/21  (A)
Substitute the value of x in (5)
5(58/21)  2y = 12
⇒ 290/21  2y = 12
⇒ 2y = 290/21  12
= (290  252)/21
= 38/21
⇒ y = 19/21  (B)
Substituting the values of x and y in z = 2  2x + 4y, we have
z = 2  2(58/21) + 4(19/21)
= (42  116 + 76)/21
= 2/21  (C)
From (A), (B), (C), we have x = 58/21, y = 19/21, and z = 2/21
Answer: Solution is x = 58/21, y = 19/21, and z = 2/21.

Example 3: Find the solution of simultaneous equations x  y = 10 and 2x + y = 9.
Solution: We will solve the given equations using the elimination method.
Adding x  y = 10 and 2x + y = 9, we have
(x  y) + (2x + y) = 10 + 9
⇒ x + 2x  y + y = 19
⇒ 3x = 19
⇒ x = 19/3
So, we have
19/3  y = 10
⇒ y = 19/3  10
= (19  30)/3
= 11/3
Answer: The solution of x  y = 10 and 2x + y = 9 is x = 19/3 and y = 11/3.
FAQs on Simultaneous Equations
What are Simultaneous Equations?
Simultaneous equations are two or more algebraic equations that share common variables and are solved at the same time (that is, simultaneously).
How to Solve Simultaneous Equations?
We use different methods to solve simultaneous equations. Some of the common methods are:
 Substitution Method
 Elimination Method
 Graphical Method
What is the Substitution Method in Simultaneous Equations?
According to the substitution method, we obtain the value of one variable in terms of another and then substitute that into another equation to find the value of the other variable.
What is the Rule for Simultaneous Equations?
Some of the important rules of simultaneous equations are:
 Simplify each side of the equation first by removing the parentheses, if any.
 Combine the like terms.
 Isolate the variable terms on one side of the equation.
 Then, use the appropriate method to solve for the variable.
What are Linear Simultaneous Equations?
Linear simultaneous equations refer to simultaneous equations where the degree of the variables is one.
How to Solve 3 Simultaneous Equations?
We can solve 3 simultaneous equations using various methods such as:
 Substitution Method
 Elimination Method
 Graphical Method
It also depends upon the number of variables involved.
What are the Three Methods to Solve Simultaneous Equations?
The three methods to solve simultaneous equations are:
 Substitution Method
 Elimination Method
 Graphical Method
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