Section 2
The existence of a unique solution
A set of linear simultaneous equations may have (i) a unique solution, (ii) no solution, or (iii) infinitely many solutions.
In a set of linear simultaneous equations, a unique solution exists if and only if, (a) the number of unknowns and the number of equations are equal, (b) all equations are consistent, and (c) there is no linear dependence between any two or more equations, that is, all equations are independent.
In a system of linear simultaneous equations if one or more equations are inconsistent, the system does not have any solution. For example, if in a set of linear simultaneous equations with two equations and two unknowns, one equation is \(x+y=2\) and another equation is \(3x+3y=5\), these two equations are inconsistent within the given system. They are inconsistent because if \(x+y=2\), then \(3x+3y\) must be \(6\), not \(5\). One cannot solve the system of linear simultaneous equations \(x+y=2\) and \(3x+3y=5\) as they are inconsistent.
Graphically, the solution of two linear simultaneous equations in two unknowns is equivalent to finding where the lines of the two equations cross. If these two equations are inconsistent, corresponding lines in the Cartesian plane are parallel and will never cross (See Practice Question 2).
In a system of linear simultaneous equations if all equations are consistent, but (a) the number of independent equations is less than the number of unknowns, and/or (b) there exists a linear dependence between two or more equations in the system, there may exists infinitely many solutions that satisfy the system.
Linear dependence, for example between two linear equations, refers to a situation when one equation in the system is a multiple of another equation. For example, equations \(y=x+2\) and \(2y=2x+4\) are linearly dependent as the later can be obtained by multiplying the former equation by \(2\).
Consider a more general example. Suppose two linear simultaneous equations are:$$a_{11}x_{1} + a_{12}x_{2} = b_{1}$$$$a_{21}x_{1} + a_{22}x_{2} = b_{2}$$Where, \(a_{ij}\) is the coefficient, \(x_j\) is the variable, and \(b_i\) is the constant. In this system if \(a_{1j}=ka_{2j}\) and \(b_1=kb_2\), where \(k\) is a constant, the equations are linearly dependent. Graphically the lines representing the graphs of two equations coincide if the equations are linearly dependent, and every point on either line is a solution. [ See Practice Question 3]
One interesting form of linear dependence may arise in a system of \(m×n\) linear simultaneous equations when one equation is the sum or difference of more than one equation in the system. For example, equations (i) \(x+y+z=10\),(ii) \(2x-2y-2z=4\), and (iii) \(3x-y-z=14\) have linear dependence. (Why?)
To sum up, consider a system of linear simultaneous equations where all equations are consistent, however, due to the linear dependence between some equations the number of independent equations is less than the number of unknowns. Such a system has infinitely many solutions.
It follows from the discussion in this section is that two linear simultaneous equations in two unknowns can have a unique solution, no solution or infinitely many solutions and this is true for every system of linear simultaneous equations with \(m\) equations and \(n\) unknowns. To read more about the existence of a unique solution, inconsistency, and linear dependence, please see the recommended books.
FAQs
The existence of a unique solution
For example, if in a set of linear simultaneous equations with two equations and two unknowns, one equation is x+y=2 x + y = 2 and another equation is 3x+3y=5 3 x + 3 y = 5 , these two equations are inconsistent within the given system.
Which two equations have unique solutions? ›
Hence, the equations a 1 x + b 1 y + c 1 = 0 , a 2 x + b 2 y + c 2 = 0 have a unique solution if a 1 a 2 ≠ b 1 b 2 .
What is a one unique solution? ›
A unique solution means only one solution. If a linear equation has a unique solution means only one solution set exists for the equation. A system of linear equations a 1 x + b 1 y = 0 a 2 x + b 2 y = 0 has a unique solution, if a 1 a 2 ≠ b 1 b 2 .
What is a unique solution of two lines? ›
UNIQUE SOLUTION:
Two lines a1 + b1y + c1 = 0 and a2x + b0y + c2 = 0, if the denominator a1b2 – a2b1 ≠ 0 then the given system of equations has unique solution (i.e. only one solution) and the solutions are said to be consistent.
What is a unique real solution? ›
A unique solution means there is a solution, but only one of them. For example, the equation x2=4 x 2 = 4 has a solution, namely x=−2 , but it is not unique since x=2 is another solution.
What are common examples of solutions? ›
Examples of Solutions
Sugar-water, salt solution, brass, alloys, alcohol in water, aerosol, air, aerated drinks such as Coca-Cola etc. are examples of solutions. When we work with chemistry, we generally prepare many types of solutions such as copper in water, iodine in alcohol etc.
What equation has two distinct real solutions? ›
The quadratic equation ax2+bx+c=0 a x 2 + b x + c = 0 will have two distinct real solutions if D=b2−4ac D = b 2 − 4 a c will be >0 . If D=0 , the quadratic equation has 1 double solution. And if D<0 , the quadratic equation has 2 distinct complex solution. This is a theory behind the quadratic equation.
Which equation has two solutions? ›
A quadratic equation with real or complex coefficients has two solutions, called roots.
What is one solution example? ›
Basically, an equation can have: Exactly one solution, like 2x = 6. It solves as x = 3, no other options. No solutions, like x+6 = x+9.
What is no unique solution? ›
To be clear, for a linear system represented by Ax=b, if there is a unique solution then A is invertible and the solution is given formally by. x=A−1b. If there is not a unique solution, then A is not invertible. We then say that the matrix A is singular.
If your system has a unique solution, then everything intersects at exactly one point. In other words, only one point satisfies each and every equation.
What is the uniqueness of your solution? ›
By the term unique solution, one mean to say that only one specific solution set exists for a given equation. What this pretty much means is, depending upon how many equations we have, all the equations will intersect at one particular point.
What is a unique solution with example? ›
The unique solution of a linear equation means that there exists only one point, on substituting which, L.H.S and R.H.S of an equation become equal. The linear equation in one variable has always a unique solution. For example, 3m =6 has a unique solution m = 2 for which L.H.S = R.H.S.
Is unique solution consistent? ›
i) If both the lines intersect at a point, then there exists a unique solution to the pair of linear equations. In such a case, the pair of linear equations is said to be consistent.
Can two planes have a unique solution? ›
The system of equations corresponding to the intersection of two planes will have either zero solutions or an infinite number of solutions. It is not possible for two planes to intersect at a single point.
How to know if a solution is unique? ›
A nxn nonhom*ogeneous system of linear equations has a unique non-trivial solution if and only if its determinant is non-zero. If this determinant is zero, then the system has either no nontrivial solutions or an infinite number of solutions.
What is the uniqueness of the solution? ›
Solutions are only guaranteed to exist locally. Uniqueness is especially important when it comes to finding equilibrium solutions. Uniqueness of solutions tells us that the integral curves for a differential equation cannot cross.
Which is the best example of a solution? ›
Some examples of solutions are salt water, rubbing alcohol, and sugar dissolved in water. When you look closely, upon mixing salt with water, you can't see the salt particles anymore, making this a hom*ogeneous mixture. Let's make use of our salt water example to talk about the two main parts of a solution.
What is an example of a one distinct real solution? ›
In the case of one real solution, the value of discriminant b2 - 4ac is zero. For example, x2 + 2x + 1 = 0 has only one solution x = -1.
