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Linear Equations in Two Variables
A linear equation in two variables is an equation in which two variables have the exponent 1. A system of equations with two variables has a unique solution, no solutions, or infinitely many solutions. 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 that there must be n equations to solve and determine the value of variables.
Linear equations in two variables are the algebraic equations which are of the form (or can be converted to the form) y = mx + b, where m is the slope and b is the yintercept. They are the equations of the first order. For example, y = 2x + 3 and 2y = 4x + 9 are twovariable linear equations.
1.  What are Linear Equations in Two Variables? 
2.  Forms of Linear Equations in Two Variables 
3.  Solving Pairs of Linear Equations in Two Variables 
4.  FAQs on Linear Equations in Two Variables 
What are Linear Equations in Two Variables?
The linear equations in two variables are the equations in which each of the two variables is of the highest order (exponent) of 1 and may have one, none, or infinitely many solutions. The standard form of a twovariable linear equation is ax + by + c = 0 where x and y are the two variables. The solutions can also be written in ordered pairs like (x, y). The graphical representation of the pairs of linear equations in two variables includes two straight lines which could be:
Forms of Linear Equations in Two Variables
A linear equation in two variables can be in different forms like standard form, intercept form and pointslope form. For example, the same equation 2x + 3y=9 can be represented in each of the forms like 2x + 3y  9=0 (standard form), y = (2/3)x + 3 (slopeintercept form), and y  5/3 = 2/3(x + (2)) (pointslope form). Look at the image given below showing all these three forms of representing linear equations in two variables with examples.
The system of equations means the collection of equations and they are also referred to as simultaneous linear equations. We will learn how to solve pair of linear equations in two variables using different methods.
Solving Pairs of Linear Equations in Two Variables
There are five methods to solve pairs of linear equations in two variables as shown below:
 Graphical Method
 Substitution Method
 Cross Multiplication Method
 Elimination Method
 Determinant Method
Graphical Method for Solving Linear Equations in Two Variables
The steps to solve linear equations in two variables graphically are given below:
 Step 1: To solve a system of two equations in two variables graphically, we graph each equation. To know how, click here or follow steps 2 and 3 below.
 Step 2: To graph an equation manually, first convert it to the form y = mx+b by solving the equation for y.
 Step 3: Start putting the values of x as 0, 1, 2, and so on and find the corresponding values of y, or viceversa.
 Step 4: Identify the point where both lines meet.
 Step 5: 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 may not intersect always. Sometimes they may be parallel. In that case, the pairs of linear equations in two variables have 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.
Consistent and Inconsistent System of Linear Equations:
 If the system has a solution, then it is said to be consistent;
 otherwise, it is said to be inconsistent.
Independent and Dependent System of Linear Equations:
 If the system has a unique solution, then it is independent.
 If it has an infinite number of solutions, then it is dependent. It means that one variable depends on the other.
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. Here we can understand when a linear system with two variables is consistent/inconsistent and independent/dependent.
Method of Substitution
To solve a system of two linear equations in two variables using the substitution method, we have to use the steps given below:
 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:
2x5y+13=0
⇒ 2x5((7x)/2)+13=0
⇒ 2x(35/2)+(5x/2)+13=0
⇒ 2x + (5x/2) = 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 crossmultiplying 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 linear equations in two variables using the elimination method, we will use the steps given below:
 Step 1: Arrange the equations in the standard form: ax+by+c=0 or ax+by=c.
 Step 2: Check if adding or subtracting the equations would result in the cancellation of a variable.
 Step 3: If not, multiply one or both equations by either the coefficient of x or y such that their addition or subtraction would result in the cancellation of any one of the variables.
 Step 4: Solve the resulting single variable equation.
 Step 5: Substitute it in any of the given 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
Adding or subtracting 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 6 and 6 such 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,
⇒ 5y15=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 × 2 matrix is obtained by crossmultiplying elements starting from the top left corner and subtracting the products.
Consider a system of linear equations in two variables: 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), follow the steps given below:
 Step 1: We first find the determinant formed by the coefficients of x and y and label it Δ.
Δ = \(\left\begin{array}{ll}a_1 & b_1 \\a_2 & b_2\end{array}\right = a_1 b_2  a_2b_1\)  Step 2: Then we find the determinant Δ_{x} which is obtained by replacing the first column of Δ with constants.
Δ_{x} = \(\left\begin{array}{ll}c_1 & b_1 \\c_2 & b_2\end{array}\right = c_1 b_2  c_2b_1\)  Step 3: We then find the determinant Δ_{y} which is obtained by replacing the second column of Δ with constants.
Δ_{y} = \(\left\begin{array}{ll}a_1 & c_1 \\a_2 & c_2\end{array}\right = a_1 c_2  a_2c_1\)
Now, the solution of the given system of linear equations is obtained by the formulas:
 x = Δ_{x} / Δ
 y = Δ_{y} / Δ
Important Points on Linear Equations with Two Variables:
 A linear equation in two variables is of the form ax + by + c = 0, where x and y are variables; and a, b, and c are real numbers.
 A pair of linear equations are of the form a_{1}x + b_{1}y + c_{1} = 0 and a_{2}x + b_{2}y + c_{2} = 0 and its solution is a pair of values (x, y) that satisfy both equations.
 To solve linear equations in two variables, we must have at least two equations.
 A linear equation in two variables has infinitely many solutions.
Tricks and Tips:
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.
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Linear Equations in Two Variables 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.
And the number when the digits are reversed is 10y+x.
The question says, "The sum of the digits of a twodigit number is 8".
So from this, we get a linear equation in two variables: 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 (by substituting the value of y)
⇒ 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
Answer: The required number is 35. 
Example 2: Jake's piggy bank has 11 coins (only quarters or dimes) that have a total value of $1.85. How many dimes and quarters does the piggy bank has?
Solution: Let us assume that the number of dimes be x and the number of quarters be y in the piggy bank. Let us form linear equations in two variables based on the given information.
Since there are 11 coins in total, x+y=11 ⇒ y=11x.
We know that, 1 dime = 10 cents and 1 quarter = 25 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 value of x in x+y=11.
⇒ y=116=5Answer: 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. Find,
 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 the 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 upstream = (xy). During downstream, the current's speed adds to the boat's speed and the speed of the boat 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, we can form a pair of linear equations in two variables:
xy=15
x+y=17Adding both equations we get:
2x = 32
⇒ x=16Substitute x=16 in x+y=17
16+y= 17
y=1Answer: Therefore, the speed of the boat is 16 miles per hour and the speed of the current is 1 mile per hour.
FAQs on Linear Equations in Two Variables
What is Meant By Linear Equation in Two Variables?
A linear equation is an equation with degree 1. A linear equation in two variables is a type of linear equation in which there are 2 variables present. For example, 2x  y = 45, x+y =35, ab = 45 etc.
How do you Identify Linear Equations in Two Variables?
We can identify a linear equation in two variables if it can be 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 a Pair of Linear Equations in Two Variables?
Yes, we can solve pair of linear equations in two variables using different methods and ensure there are two equations present in the given system of equations so as to obtain the values of variables. If there is one solution it means that the given lines are intersecting, if there is no solution possible, then it means that the given equations are of parallel lines. If there are infinitely many solutions possible, it means that the given equations are forming coincidental lines.
How to Graphically Represent a Pair of Linear Equations in Two Variables?
We can represent linear equations in two variables graphically using the steps given below:
 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 pair of linear equations in two variables.
How Does One Solve the System of Linear Equations in Two Variables?
We have different methods to solve the system of linear equations:
 Graphical Method
 Substitution Method
 Cross Multiplication Method
 Elimination Method
 Determinant or Matrix 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 a linear equation in two variables have the following things in common:
 The degree of 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 is separated by an inequality symbol. But note that the inequality rules should be taken care of.
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