System of Equations
In mathematics, a system of equations, also known as a set of simultaneous or an equation system, is a finite set of equations for which we sought the common solutions. A system of equations can be classified in a similar manner as single equations. System of equations finds applications in our daytoday lives in modeling problems where the unknown values can be represented in form of variables.
What is a System of Equations?
In algebra, a system of equations comprises two or more equations and seeks common solutions to the equations. "A system of linear equations is a set of equations which are satisfied by the same set of variables."
System of Equations Example
A system of equations as discussed above is a set of equations that seek a common solution for the variables included. The following set of equations is an example of system of equations,
 2x  y = 12
 x  2y = 48
Solutions to System of Equations
Solving a system of equations means finding the values of the variables used in the set of equations. We compute the values of the unknown variables still balancing the equations on both sides. The main reason behind solving an equation system is to find the value of the variable that satisfies the condition of all the given equations true. There can be different types of solutions to a given system of equations,
 Unique solution
 No solution
 Infinitely many solutions
Unique Solution of a System of Equations
The unique solution of a system of equations means that there exists only one value for the variable or the point of intersection of the lines representing those equations, on substituting which, L.H.S and R.H.S of all the given equations in the system become equal.
For example, we know that a linear equation in one variable will always have one solution. Let understand the concept of a unique solution using a linear equation in one variable, 4x = 8 has a unique solution x = 2 for which the L.H.S is equal to the R.H.S.
Similarly, for a system of linear equations in two variables, the unique solution is an ordered pair (x, y) which will satisfy both the equations in the system.
No Solution
A system of equations has no solution when there exists no point where lines intersect each other or the graphs of equations are parallel.
Infinite Many Solutions
A system of equations can have infinitely many solutions when there exists a solution set of infinite points for which L.H.S and R.H.S of an equation become equal, or in the graph straight lines overlap each other.
Solving System of Equations
Any system of equations can be solved in different methods. To solve a system of equations in 2 variables, we need at least 2 equations. Similarly, for solving a system of equations in 3 variables, we will require at least 3 equations. Let us understand 3 ways to solve a system of equations given the equations are linear equations in two variables.
 Substitution Method
 Elimination Method
 Graphical Method
Solving System of Equations Using Substitution Method
For solving the system of equations using the substitution method given two linear equations in x and y, express y in terms x in one of the equations and then substitute it in 2nd equation. Consider
3x − y = 23 → (1)
4x + 3y = 48 → (2)
From(1), we get:
y = 3x − 23 → 3
Plug in y in (2),
4x + 3 (3x − 23) = 48
13x − 69 = 48
13x = 117
⇒x = 9
Now, plug in x = 9 in (1)
y = 3 × 9 − 23 = 4
Hence, x = 9 and y = 4 is the solution of given system of equations.
Solving System of Equations Using Elimination Method
Using the elimination method to solve the system of equations, we eliminate one of the unknowns, by multiplying equations by suitable numbers, so as the coefficients of one of the variables become the same. Consider
2x + 3y = 4 → (1) and 3x + 2y = 11 → (2)
The coefficients of y are 3 and 2; LCM (3, 2) = 6
Multiplying Eqn(1) by 2 and Eqn(2) by 3, we get
4x + 6y = 8 → (3)
9x + 6y = 33 → (4)
On subtracting (3) from (4), we get
5x = 25
⇒x = 5
Plugging in x = 5 in (2) we get
15 + 2y = 11
⇒y = −2
Hence, x = 5, y = −2 is the solution.
Solving System of Equations Using Graphical Method
In this method, the solution of simultaneous equations is obtained by plotting their graphs. "The point of intersection of the two lines is the solution of the system of equations using graphical method."
Example: 3x + 4y = 11 and x + 2y = 3
Find at least two values of x and y satisfying equation 3x + 4y = 11
x  1  3 

y  2  0.5 
So we have 2 points A (1,2 ) and B (3,(1/2)).
Similarly, find the at least two values of x and y satisfying equation x + 2y = 3
x  3  3 

y  0  3 
We have two points C(3, 0) and D( 3, 3).
Plotting these points on the graph we can get the lines in a coordinate plane as shown below.
We observe that the two lines intersect at (1,2). So, x = 1, y = 2 is the solution of given system of equations. Methods I and II are the algebraic way of solving simultaneous equations and III is the graphical method.
Solving System of Equations Using Matrices
The solution of a system of equations can be solved using matrices. In order to solve a linear equation using matrices, express the given equations in standard form, with the variables and constants on respective sides. for the given equations,
a\(_1\)x + \(b_1\)y + \(c_1\)z = \(d_1\)
a\(_2\)x + \(b_2\)y + \(c_2\)z = \(d_2\)
a\(_3\)x + \(b_3\)y + \(c_3\)z = \(d_3\)
we can express them in the form of matrices as,
\(\left[\begin{array}{ccc}
a_1x + b_1y + c_1 z \\
a_2x + b_2y + c_2 z \\
a_3x + b_3y + c_3 z
\end{array}\right] = \left[\begin{array}{ccc}
d_1 \\
d_2 \\
d_3
\end{array}\right]\)
⇒\(\left[\begin{array}{ccc}
a_1 & b_1 & c_1 \\
a_2 & b_2 & c_2 \\
a_3 & b_3 & c_3
\end{array}\right] + \left[\begin{array}{ccc}
x \\
y \\
z
\end{array}\right] = \left[\begin{array}{ccc}
d_1 \\
d_2 \\
d_3
\end{array}\right]\)
⇒ AX = B
Here,
A = \(\left[\begin{array}{ccc}
a_1 & b_1 & c_1 \\
a_2 & b_2 & c_2 \\
a_3 & b_3 & c_3
\end{array}\right]\), X = \(\left[\begin{array}{ccc}
x \\
y \\
z
\end{array}\right]\), B = \(\left[\begin{array}{ccc}
d_1 \\
d_2 \\
d_3
\end{array}\right]\)
⇒ X = A^{1}B
Applications of System of Equations
Systems of equations are a very useful tool and find application in our daytoday lives for modeling reallife situations and analyzing questions about them. For applying the concept of the system of equations, we need to translate the given situation into two linear equations with two variables, then further solve to find the solution of linear programming problems. Any method to solve the system of equations, substitution, elimination, graphical, etc methods. Follow the belowgiven steps to apply the system of equations to solve problems in our daily lives,
 To translate and represent the given situation in form of a system of equations, identify unknown quantities in a problem represent them with variables.
 Write a system of equations modeling the conditions of the problem.
 Solve the system of equations.
 Check and express the obtained solution.
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Check these articles related to the concept of the system of equations.
System of Equations Examples

Example 1: In a ΔLMN, ∠N = 3∠M = 2(∠L+∠M). Calculate the angles of ΔLMN using this information.
Solution:
Let \(\angle L = x^{\circ} \,and\, \angle M =y^{\circ}.\)
Then,
\(\angle N = 3\angle M =(3y)^{\circ}.\)
Now,
\(\begin{align} \angle L+\angle M+\angle N&=180 ^{\circ}\\\therefore x + y + 3y&=180\\x + 4y&=180\,...\,\,(1)\end{align}\)
Also, \(\begin{align} \,\,\angle N&=2(\angle L+\angle M) \\\therefore 3y&=2(x+y)\\\Rightarrow2xy&=0\,...\,\,(2)\end{align}\)
On multiplying (2) by 4 and adding the result to (1), we get \(\begin{align} 9x&=180\\x&=20. \end{align}\)
Putting \( x=20\) in (2), we get \( y=40\) and \(3y = 120\)
\(\therefore \angle L= 20 ^{\circ} , \angle M = 40^{\circ} \,and \,\angle N=120^{\circ}\)

Example 2: Peter is three times as old as his son. 5 years later, he shall be two and half times as old as his son. What's Peter's present age?
Solution:
Let Peter's age be \(x\,\) years and his son's age be \(y\,\)years.
\(x=3y\,...\,Eqn(1)\)
5 years later,
Peter's age \(=x+5\,\) years and his son's age \(=y+5\,\)years.
\(\begin{align} x+5&=\dfrac{5}{2}(y+5)\\2x5y15&=0 \,...\,Eqn(2)\end{align}\)
Put \(x=3y\) in Eqn(2), we get
\(\begin{align}6y&=5y+15\\\therefore y&=15 \end{align}\)
Plug in \(y=15\) in Eqn(1), we get
\(x=45\)
\(\therefore\) Present age of Peter = 45 years and Present age of son= 15 years

Example 3: Tressa starts her job with a certain monthly salary and earns a fixed increment every year. If her salary was $1500 after 4 years and $1800 after 10 years of service, find her starting salary and annual increment.
Solution:
Let her starting salary be \(x\) and annual increment be \(y\).
After 4 years, her salary was $1500
\(x+4y=1500\,...\,Eqn\,(1)\)
After 10 years, her salary was $1800
\(x+10y=1800\,...\,Eqn\,(2)\)
Subtracting Eqn (1) from (2), we get
\(\begin{align} 6y&=300\\\therefore y&=50\end{align}\)
On putting \(y=50\) in Eqn (1), we get \(x=1300\)
\(\therefore\)Starting salary was $1300 and annual increment is $50
FAQs on System of Equations
What is a System of Equations in Mathematics?
A system of equations in mathematics, also known as a set of simultaneous or equation system, is a finite set of equations for which we sought the common solutions.
How Do You Solve a System of Equations?
Solving a system of equations is computing the unknown variables still balancing the equations on both sides. We solve an equation system is to find the value of the variable that satisfies the condition of all the given equations true. There are different methods to solve a system of equations,
 Graphical method
 Substitution method
 Elimination method
 Crossmultiplication method
How Do You Create a System of Equations with Two Variables?
To create a system of equations with two variables:
 First, identify the two unknown quantities in the given problem.
 Next, find out the two conditions given and frame equations for each of them.
With these two steps, a system of equations with two variables gets created.
How to Solve a System of Equations Using Substitution Method?
The substitution method is one of the ways to solve a system of equations in two variables, given the set of linear equations. In this method, we substitute the value of a variable found by one equation in the second equation.
How to Solve a System of Equations Using Elimination Method?
The elimination method is used to solve a system of equations that are linear. In the elimination method, we eliminate one of the two variables and try to solve equations with one variable.
How to Solve a System of Equations With Graphical Method?
To solve the system of equations, given a set of linear equations graphically, we need to find at least two solutions for the respective equations. We observe the pattern of lines after plotting the point to infer it is consistent, dependent, or inconsistent. If the two lines are intersecting at the same point, then the point gives a unique solution for the system of equations. If the two lines coincide, then in this case there are infinitely many solutions and if the two lines are parallel, then in this case there is no solution.
How to Solve a System of Equations With Cross Multiplication Method?
While solving a system of equations using the cross multiplication method, the numerator of one fraction is multiplied to the denominator of another and the denominator of the first term to the numerator of another term.
How do you Solve a System of Equations Using 2 Equations with 3 Variables?
An equation with 3 variables represents a plane.
 Step 1) To solve a system of 2 equations with 3 variables say x, y, and z, we will consider the 1st two equations and eliminate one of the variables, say x, to obtain a new equation.
 Step 2) Next, we write the 2nd variable, y in terms of z from the new equation and substitute it in the third equation.
 Step 3) Assuming z = a, we will obtain values of x and y also in terms of a.
 Step 4) Once, we know the value of a, we can find the values of x, y, and z.
How Many Ways Can You Solve a System of Equations?
There are three main methods to solving system of equations, they are:
 Substitution Method
 Elimination Method
 Graphical Method
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