Question

How can you tell if an equation has infinitely many solutions?

Answer 1

Hint: We will use the basic definition of the equation and then the properties of the equation and use all the conditions to explain the infinite solution of the equation. The number of solutions of an equation usually depends on the number of variables. The term “infinite” signifies unboundedness or limitlessness.

Complete step by step solution:
Here we need to explain the conditions for an equation to have infinitely many solutions.
If the equation ends with a true statement then the given equation will have infinitely many solutions. For example: \[2x + 2 = 2\left( {x + 1} \right)\]. Here we can see that the equation ends with a true statement as every value of the variable satisfies the equation.
If the equation has more than one variable and also it does not force uniqueness, then the equation will have infinitely many solutions.
For example: the equation \[{x^2} + {y^2} = 1\] has infinitely many solutions, but the equation \[{x^2} + {y^2} = 0\] has only one solution.
In a graph, the system of equations is said to have infinitely many solutions, only when the lines are coincident, and they have the same \[y\]-intercept.
Hence, these are the conditions for an equation to have infinitely many solutions.

Note:
An equation is defined as a mathematical statement that consists of an equal symbol between two algebraic expressions that have the same value. We need to keep in mind that it may be extremely difficult to determine the number of solutions in the case of Diophantine equations which are defined as the equations where the values of the variables are limited to integers or to positive integers.