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.
The set of solutions obtained by solving these linear equations is a straight line. 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.
The linear equations in two variables are the equations in which each of the two variables are of the highest exponent order of 1 and 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 system of linear equations in two variables includes two straight lines which could be intersecting lines, parallel lines, or coincident lines.
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 referred to as simultaneous linear equations. We will learn how to solve linear equations in two variables using different methods.
There are five methods to solve a system of linear equations in two variables. Those methods are explained below:
The steps to solve linear equations in two variables graphically are given below:
Example: Find the solution of the following system of equations graphically.
x + 2y  3 = 0
3x + 4y  11= 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 system of linear equations in two variables 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.
Consistent and Inconsistent System of Linear Equations:
Independent and Dependent System of Linear Equations:
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.
To solve a system of two linear equations in two variables using the substitution method, we have to use the steps given below:
Example: Solve the following system of equations using the substitution method.
x + 2y7 = 0
2x  5y + 13 = 0
Solution: Let us solve the equation, x + 2y  7 = 0 for y:
x + 2y  7 = 0
⇒2y = 7  x
⇒ y=(7  x)/2
Substitute this in the equation, 2x  5y + 13 = 0:
2x  5y + 13 = 0
⇒ 2x  5((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=(7  1)/2 = 3
Therefore, the solution of the given system is x = 1 and y = 3.
Equation 1: 3x  y = 7
Equation 2: 2x + 3y = 12
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.
From this equation, we get two equations:
Solving each of these for x and y, the solution of the given system is:
To solve a system of linear equations in two variables using the elimination method, we will use the steps given below:
Example: Solve the following system of equations using the elimination method.
2x + 3y 11= 0
3x + 2y  9 = 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 + 3y  11 = 0)
⇒ 6x + 9y  33 = 0
2 × (3x + 2y  9 = 0)
⇒ 6x  4y + 18 = 0
Now we will add these two equations:
6x + 9y  33 = 0
6x  4y + 18 = 0
On adding both the above equations we get,
⇒ 5y  15 = 0
⇒ 5y = 15
⇒ y = 3
Substitute this in one of the given two equations and solve the resultant variable for x.
2x + 3y  11 = 0
⇒ 2x + 3(3)  11 = 0
⇒ 2x + 9  11 = 0
⇒ 2x = 2
⇒ x = 1
Therefore, the solution of the given system of equations is x = 1 and y = 3.
Equation 1: 4x  2y = 10
Equation 2: 3x + y = 5
Which of the following choices presents the correct values of 'x' and 'y' that satisfy the system of equations?
The determinant of a 2 × 2 matrix is obtained by cross multiplying 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:
Now, the solution of the given system of linear equations is obtained by the formulas:
x = Δ_{x }/ Δ
y = Δ_{y} / Δ
Tricks and Tips on Linear Equations with Two Variables:
While solving the equations using either the substitution method or the elimination method:
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 = 8  x
Also, when the digits are reversed, the number is increased by 18.
So, the equation is 10y + x =10x + y + 18
⇒ 10(8  x) + x =10x + (8  x) +18 (by substituting the value of y)
⇒ 80  10x + x =10x + 8  x + 18
⇒ 80  9x = 9x + 26
⇒ 18x = 54
⇒ x = 3
Substituting x=3 in y = 8  x, we get,
⇒ y = 8  3 = 5
⇒ 10x + y = 10(3) + 5 = 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(11  x) = 185 (as y = 11x)
⇒ 10x + 275  25x =185
⇒ 15x +275 =185
⇒ 15x = 90
⇒ x = 6
Substitute this value of x in x + y =11.
⇒ y = 11  6 = 5
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