Consistent Linear Equations

In mathematics, linear equations occur when the value of one variable or two variables in a relation is set and the other must be found. Solving linear equations undergo the various rules like addition and subtraction, we can multiply both sides of an equation by a number without changing the equation.
Sometime we identifies variable is a part of any fraction, we must avoid values which make denominator of the fraction zero. In this section, we will learn about when the system of linear equations called consistent with the help of some solved examples.

Consistent Solution Linear Equations

In mathematics, mainly in linear algebra, linear or nonlinear system of equations is consistent if it has any set of solutions. Solving a linear equation usually means solving the system fro unknown values. Linear equations are consistent if they contains at least one set of values for the unknowns variables which satisfy all the equations in the system.

How do you determine if a system is consistent or inconsistent?
If any system of equation is consistent either it has one solution or infinitely many solutions.

 System of equations has exactly one solution Consistent System System is Independent System has an infinite number of solutions Consistent System System is Dependent System has no solution Inconstant System

After comparing the coefficients of linear equations or after solving linear equation, we can tell if the system of linear equations is consistent or inconsistent.

Problems on Linear Equations

Example 1: Solve 2(m - 2) = 10
Solution: 2(m - 2) = 10
2m - 4 = 10
2m = 10 + 4
2m = 14
m = 7
The value of m is 7.
Verification: 2(7 - 2) = 10
2(5) = 10
10 = 10 (RHS = LHS)
Which is true.
Example 2: Are coincident lines consistent or inconsistent?
Solution:  Coincident lines are consistent. Will take an example of linear equations with two variables to justify, say 2x - y =0 or y = 2x. The graph of this equation is a straight line. The statement will be true for all the values.
We can conclude that, the system has an infinite many solutions.
So all the sets having coincident lines are consistent-dependent systems.
Whereas, a system with no solutions is an inconsistent system.

Example 3: Show that if below system of equations has consistent solution.
2x - y = -5 and -2x = 2y - 4

Solution: 2x - y = -5 and -2x = 2y - 4
Arrange the equations as
2x - y = -5 and x + y = 2
Plot a graph, If lines intersect or coincide, then only system of equations has consistent solution. Since graph intersect at point (-1, 3), which is solution of the given system. This implies system is consistent.

Practice Problems

Practice Problem 1: Solve 4(m - 12) = 3(m - 4)
Practice Problem 2: Verify if below systems are consistent or inconsistent.
a) x + y = 2 and x - y = 10
b) 2x - 4y = 9 and 5x - y = 7
c) x = 3 and y - 7 = 10y
Practice Problem 3: How many solutions does a pair of 2 equations with two variables has if lines intersect at one point? Justify your answer with any example.