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If the process of substituting leads to an identity rather than an equation then the system has infinitely many solutions.

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Q: When you use the substitution method how can you tell that a has an infinite number of solutions?
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What is the number of solutions of a system in which the lines are coincident?

an infinite number of solutions


When using the addition or substitution method how can you tell whether a system of linear equations has infinitely many solutions and what is the relationship between the graphs of the two equation?

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It depends on the equation. Also, the domain must be such that is supports an infinite number of solutions. A quadratic equation, for example, has no real solution if its discriminant is negative. It cannot have an infinite number of solutions. Many trigonometric equations are periodic and consequently have an infinite number of solutions - provided the domain is also infinite. A function defined as follows: f(x) = 1 if x is real f(x) = 0 if x is not real has no real solutions but an infinite number of solutions in complex numbers.


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How many solutions do these equations havey = 2x + 42y = 4x + 8?

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If a system has an infinite number of solutions does it follow that any ordered pair is a solution?

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