In linear algebra, linear equations is one of the most important topics. We deal with various linear equations with number of variables such as linear equations with one variable, linear equations with two variable, linear equations with three variable etc. Solving linear equations means to find the solutions for a particular given set of linear system which must satisfy all of the equations. Solution for linear equation in one variable is a single point, or a line or the empty set. While dealing with set of two linear equations, we have different sets of solutions like: unique solution, no solution or infinite many solutions.

Solving Systems of Linear Equations

What is the solution of a system of linear equations? A system of linear equations is a set of two or more linear equations. The pair of 2 linear equations involves two variables with degree one. Each linear equation is a straight line on the coordinate plane (xy-plane). Any set of solutions which satisfy the given equations is the solution set for the given equations.

Algebraic and Graphical, two ways to get the solution set for the system of linear equations.

Graphing method, Substitution Method, Matrix method and Elimination method are the most important algebraic methods.

Whereas, in the graphical method, we get 3 possibilities:

The lines are parallel (No Solution)
Lines intersect at one point. (Unique Solution)
All equations represent same line ( Infinitely many Solutions)

Questions On Linear Equations

Question 1: Solve the following pair of equation
 
7x - 8y = 12
 
7x = 2 (6 + 4y)
 
Solution:
 
Step 1: Given equations are
7x - 8y = 12
7x = 2 (6 + 4y)

Step 2: Parenthesis is present on the right side of the second equation. Using distributive property to expand it
 
7x = 2 (6 + 4y)
 
7x = 12 + 8y
 
Step 3: The equation number 1 has its terms containing variables on one side. Similarly, we need to get the terms containing variables on one side of the another equation i.e.
 
7x = 12 + 8y
 
7x - 8y = 12
 
Step 4: Let us name the first equation as equation number 1 and the second equation as equation number 2. 
7x - 8y = 12  ...(1)
 
7x - 8y = 12 ...(2)


Step 5: Using the method of elimination to solve for the variables $x$ and $y$. We have to get the coefficient of either $x$ or $y$ same with negative sign for both the equations. So, we multiply the equation number $1$ with $-1$ throughout.
 
-1(7x - 8y = 12)
 
-7x + 8y = -12
 
Step 6: Now we add the two equations
 
7x - 8y = 12
 
-7x + 8y = -12
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0x + 0y = 0
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That is 0 = 0
 
Step 7: This results in cancellation of both the variables giving us $0$. Hence, no solution
 
Step 8: As there is no solution obtained in the given pair of equation, we can conclude that the equations are inconsistent. 

Question 2 : What are the 3 methods for solving systems of equations?
Solution: There are different ways to find the solution set of the given system of equations. The important 3 methods to solve linear equations with two variables are: Graphing method, Substitution Method and Elimination method.

Question 3
: Solve the system of linear equations by graphing.
(a) 4x - 3y = 12; x + y = 3
(b) y + 4 = -2x and y = -2x - 4

Solution:
(a)  Let us find the x and y intercepts of each equation and record it in the tables given below. 
4x - 3y = 12

 0  -4
 3  0

The coordinates of intercepts are ( 0, -4) and ( 3, 0)

x+ y = 3
 0  3
 3  0
 
The coordinates of intercepts are (0,3) and ( 3, 0)
Graphical representation: 

Solution of Linear Equations
From the graph we observe that the two lines intersect at ( 3,0) which is the solution to the above system of equations.

(b) y + 10 = -5x and y = -5x - 10

y + 10 = -5x
 x  y 
 0  -10
 -2  0
The coordinates of intercepts are (0, -10) and (-2, 0)

y = -5x - 10
 y
 0  -10
 -2  0
The coordinates of intercepts are (0, -10) and (-2, 0)

Graphical representation: 
Linear Equations with Many Solutions

From the graph we observe that, both the lines coincide. So system have infinite many solutions.

Practice Questions

Practice Question 1: Find the solution for the equation x + y = 4.
Practice Question 2: Solve 2x - 3y = 10 and 10x - 3y = 5 using elimination method.
Practice Question 3: Plot a graph for x + y = 10 and x - y = 3. Find the values which satisfy the system.
Practice Question 4: Solve -x/2 + 12(x + 3 - 2x ) + 119.