Linear Equations in Two Variables
A linear system of equations may have 'n' number of variables. An important thing to keep in mind while solving linear equations with n number of variables is, there must be n equations to solve and determine the value of variables. The set of solutions obtained on solving these linear equations is a straight line. Linear equations are the algebraic equations which are of the form y = mx + b, where m is the slope and b is y intercept. They are the equations of the first order. For example, y = 2x+3 and 2y = 4x + 9 are linear equations of 2 variables. In this minilesson, we will explore solving the system of two linear equations in two variables using different methods.
Two Variable Linear Equations
The linear equations in two variables are of the highest exponent order of 1 and have one, none or infinitely many solutions. The standard form of linear equation is ax+ by+ c= 0 where x and y are the two variables. The solutions can also be written in ordered pairs. The geometrical representation of the linear equations in two variables are also straight lines. A linear equation in two variables can be in different forms. Some of them are: standard form, intercept form and the point slope form.
The system of equations means the collection of equations. We will learn how to solve linear equations in two variables using different methods.
Methods of Solving Linear Equations in Two Variables
 Graphical Method
 Method of Substitution
 Cross Multiplication Method
 Method of Elimination
 Determinant Method
Graphical Method
The steps to solve linear equations in two variables graphically we use the below steps:
 Step 1: To solve a system of two equations in two variables graphically we graph each equation.
 Step 2: To graph an equation manually, first convert it to the form y=mx+b by solving the equation for y.
 Step 3: Identify the point where both lines meet.
 Step 4: The point of intersection is the solution of the given system.
Example: Find the solution of the following system of equations graphically.
x+2y3 =0
3x+4y11=0
Solution: We will graph them and see whether they intersect at a point.
As you can see below, both lines meet at (1, 2). Thus, the solution of the given system of linear equations is x=1 and y=2.
But both lines always may not intersect. Sometimes they may be parallel. In that case, the given system has no solution. In some other cases, both lines coincide with each other. In that case, each point on that line is a solution of the given system and hence the given system has an infinite number of solutions. If the system has a solution, then it is said to be consistent; otherwise, it is said to be inconsistent.
Consider a system of two linear equations: \(a_{1}\)x + \(b_{1}\)y + \(c_{1}\) = 0 and \(a_{2}\)x + \(b_{2}\)y + \(c_{2}\) = 0
Method of Substitution
To solve a system of two equations in 2 variables using the substitution method we use the below given steps:
 Step 1: Solve one of the equations for one variable.
 Step 2: Substitute this in the other equation to get an equation in terms of a single variable.
 Step 3: Solve it for the variable.
 Step 4: Substitute it in any of the equations to get the value of another variable.
Example: Solve the following system of equations using the substitution method.
x+2y7=0
2x5y+13=0
Solution: Let us solve the equation, x+2y7=0 for y:
x+2y7=0
⇒2y=7x
⇒ y=(7x)/2
Substitute this in the equation, 2x5y+13=0:
2 x5 y+13=0
⇒ 2x5((7x)/2)+13=0
⇒ 2x(35/2)+(5x/2)+13=0
⇒ 2x+(5x/2)=\dfrac{35}{2}13
⇒ (9x/2) = (9/2)
⇒ x=1
Substitute x=1 this in the equation y=(7x)/2:
y=(71)/2 = 3
Therefore, the solution of the given system is x=1 and y=3.
Cross Multiplication Method
Consider a system of linear equations: a_{1}x+b_{1}y+c_{1}=0 and a_{2}x+b_{2}y+c_{2}=0
To solve this using the cross multiplication method, we first write the coefficients of each of x and y and constants as follows:
Here, the arrows indicate that those coefficients have to be multiplied. Now we write the following equation by cross multiplying and subtracting the products.
\[\dfrac{x}{b_{1} c_{2}b_{2} c_{1}}=\dfrac{y}{c_{1} a_{2}c_{2} a_{1}}=\dfrac{1}{a_{1} b_{2}a_{2} b_{1}}\]
From this equation, we get two equations:
\[\begin{align}
\dfrac{x}{b_{1} c_{2}b_{2} c_{1}}&=\dfrac{1}{a_{1} b_{2}a_{2} b_{1}} \\[0.2cm] \dfrac{y}{c_{1} a_{2}c_{2} a_{1}}&=\dfrac{1}{a_{1} b_{2}a_{2} b_{1}}
\end{align}\]
Solving each of these for x and y, the solution of the given system is:
\(\begin{align}
x&=\frac{b_{1} c_{2}b_{2} c_{1}}{a_{1} b_{2}a_{2} b_{1}}\\[0.2cm] y&=\frac{c_{1} a_{2}c_{2} a_{1}}{a_{1} b_{2}a_{2} b_{1}}
\end{align}\)
Method of Elimination
To solve a system of two equations in 2 variables using the the elimination method we use the below steps:
 Step 1: If adding the equations would result in the cancellation of a variable.
 Step 2: If not, multiply one or both equations by either the coefficient of x or y such that their addition would result in the cancellation of any one of the variables.
 Step 3: Solve the resulting single variable equation.
 Step 4: Substitute it in any of the equations to get the value of another variable.
Example: Solve the following system of equations using the elimination method.
2x+3y11=0
3x+2y9=0
By adding these two equations would not result in the cancellation of any variable. Let us aim at the cancellation of x. The coefficients of x in both equations are 2 and 3. Their LCM is 6. We will make the coefficients of x in both equations to be 6 and 6 so that the x terms get canceled when we add the equations.
3 × (2x+3y11=0)
⇒ 6x+9y33=0
2 × (3x+2y9=0)
⇒ 6x4y+18=0
Now we will add these two equations:
6x+9y33=0
6x4y+18=0
On adding both the above equations we get,
⇒ 5 y15=0
⇒ 5y=15
⇒ y=3
Substitute this in one of the given two equations and solve the resultant variable for x.
2x+3y11=0
⇒ 2x+3(3)11=0
⇒ 2x+911=0
⇒ 2x=2
⇒ x=1
Therefore, the solution of the given system of equations is x=1 and y=3.
Determinant Method
The determinant of a 2 x 2 matrix is obtained by cross multiplying elements starting from the top left corner and subtracting the products.
Consider a system of linear equations: \(a_{1}\)x + \(b_{1}\)y = \(c_{1}\) and \(a_{2}\)x + \(b_{2}\)y = \(c_{2}\)
To solve them using the determinants method (which is also known as Crammer's Rule):
 We first find the determinant formed by the coefficients of x and y and label it Δ.
Δ = \[\left\begin{array}{ll}a_1 & b_1 \\a_2 & b_2\end{array}\right = a_1 b_2  a_2b_1\]  We then find the determinant \(\Delta_x\) which is obtained by replacing the first column of Δ with constants.
\(Δ_{x}\) = \[\left\begin{array}{ll}c_1 & b_1 \\c_2 & b_2\end{array}\right = c_1 b_2  c_2b_1\]  We then find the determinant \(\Delta_y\) which is obtained by replacing the second column of Δ with constants.
\(Δ_{y}\) = \[\left\begin{array}{ll}a_1 & c_1 \\a_2 & c_2\end{array}\right = a_1 c_2  a_2c_1\]
Then the solution of the given system of linear equations is obtained by the formulas:
x = \(Δ_{x}\)/Δ
y = \(Δ_{y}\)/Δ
Tips to Remember
While solving the equations using either the substitution method or the elimination method:
 If we get an equation that is true (i.e., something like 0 = 0, 1 = 1, etc), then it means that the system has an infinite number of solutions.
 If we get an equation that is false (i.e., something like 0 = 2, 3 = 1, etc), then it means that the system has no solution.
Topics Related to Linear Equations in Two Variables
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 Solutions of a Linear Equation
 Simultaneous Linear Equations
 Solving Linear Equations Calculator
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 System of Equations Solver
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 Linear Graph Calculator
 Linear Equation Formula
 Linear Equations in One Variable
 Applications of Linear Equations
Solved Examples

Example 1: The sum of the digits of a twodigit number is 8. When the digits are reversed, the number is increased by 18. Find the number.
Solution: Let us assume that x and y are the tens digit and the ones digit of the required number. Then the number is 10x+y.
The number when the digits are reversed is, 10y+x.
The problem says, "The sum of the digits of a twodigit number is 8".
So we get: x+y=8
⇒ y=8x
Also, when the digits are reversed, the number is increased by 18.
So the equation is 10y+x =10x+y+18
⇒ 10(8x)+x =10x+(8x)+18
⇒ 8010x+x =10x+8x+18
⇒ 809x=9x+26
⇒ 18x = 54
⇒ x=3Substituting x=3 in y=8x we get
⇒ y=83=5
⇒ 10x+y=10(3)+5 =35
⇒ The required number is 35. 
Example 2: Jake's piggy bank has11 coins (only quarters or dimes) that have a value of $1.85. How many dimes and quarters does the piggy bank has?
Solution: Let us assume thatthe number of dimes = x
the number of quarters = y
Since there are 11 coins in total, x+y=11 ⇒ y=11x
We know that, 1 dime= 25 cents and 1 quarter = 10 cents
The total value of the money in the piggy bank is $1.85 (185 cents).Thus we get the equation 10x + 25y = 185
⇒ 10x + 25(11x) = 185 (as y = 11x)
⇒ 10x + 275  25x =185
⇒ 15x +275 =185
⇒ 15x=90
⇒ x = 6Substitute this in x=6 in x+y=11
⇒ y=116=5
\(\therefore\) The number of dimes is 6 and the number of quarters is 5. 
Example 3: In a river, a boat can travel 30 miles upstream in 2 hours. The same boat can travel 51 miles downstream in 3 hours. Then
 What is the speed of the boat in still water?
 What is the speed of the current?
Solution: Let us assume that:
 the speed of boat in still water = x miles per hour
 the speed of current = y miles per hour.
During upstream, the current pulls back the boat's speed and the speed of the boat during upstream = (xy). During downstream, the current's speed adds to the boat's speed and the speed of the boat during downstream = (x+y).
Thus,
Distance (d)
Time (t) Speed Speed = (d/t) Upstream 30 2 (xy) 30/2=15 Downstream 51 3 (x+y) 51/3=17 Using the last two columns of the table:
xy=15
x+y=17Adding both the equations we get:
2x = 32
⇒ x=16Substitute x=16 in x+y=17
16+y= 17
y=1Therefore, the speed of boat is 16 miles per hour and the speed of current is 1 mile per hour.
FAQs on Linear Equations in Two Variables
What are Linear Equations?
A linear equation is an equation in which the variable(s) is(are) with the exponent 1. For Example, 2x = 45, x+y =35 and ab = 45.
How Do You Identify Linear Equations in Two Variables?
We can identify a linear equation in two variables if it is expressed in the form ax+by+ c = 0, consisting of two variables x and y and the highest degree of the given equation is 1.
Can You Solve an Equation with Two Variables?
Yes we can solve an equation in two variables using different methods and ensuring there are two equations present in the given system of equations so as to obtain the values of variables.
How to Graphically Represent Linear Equations in Two Variables?
We can represent linear equations in two variables using the below given steps:
 Step 1: A system of two equations in two variables can be solved graphically by graphing each equation by converting it to the form y=mx+b by solving the equation for y.
 Step 2: THe points where both lines meet are identified.
 Step 3: The point of intersection is the solution of the given system of linear equations in two variables.
How Does One Solve the System of Linear Equations?
We have different methods to solve the system of linear equations:
 Graphical Method
 Substitution Method
 Cross Multiplication Method
 Elimination Method
 Determinants Method
How Many Solutions Does a Linear Equation with Two Variables Have?
Suppose we have \(a_{1}\)x + \(b_{1}\)y + \(c_{1}\) = 0 and \(a_{2}\)x + \(b_{2}\)y + \(c_{2}\) = 0. The solutions of a linear equation with two variables are:
 One and unique if \(a_{1}\)/\(a_{2}\) ≠ \(b_{1}\)/\(b_{2}\)
 None if \(a_{1}\)/\(a_{2}\) = \(b_{1}\)/\(b_{2}\)_{ }≠ \(c_{1}\)/\(c_{2}\)
 Infinitely many if \(a_{1}\)/\(a_{2}\) = \(b_{1}\)/\(b_{2}\)_{ }= \(c_{1}\)/\(c_{2}\)
How is a Linear Inequality in Two Variables like a Linear Equation in Two Variables?
A linear inequality in two variables and linear equation in two variables have the following things common in them:
 Degree a linear equation and linear inequality is always 1.
 Both of them can be solved graphically.
 The way to solve a linear inequality is the same as linear equations except that it separated by an inequality symbol.