A pair of linear equations has infinitely many solutions when the two equations describe the same line. The "two" lines intersect everywhere.
Example
x + y = 6
2x + 2y = 12 (The coefficients are twice those in the first equation. Every solution to this equation will be a solution to the first equation.)
A pair of linear equations has no solution when they describe parallel lines. Parallel lines have no point of intersection.
Example
x + y = 6
x + y = 7 (No solution to this equation will also be a solution to the first equation)
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A single equation can have infinitely many solutions. When there are more variables than there are equations, that will generally be the case (but not always).
Example x + y = 6
A single equation may have no solutions. Usually a function is involved that has a domain or range that is not infinite.
Example x2 = -1 (The set of values produced by squaring real numbers is always positive. There does not exist a real number x that will satisfy this equation.)
Example
x + y = 6
2x + 2y = 12 (The coefficients are twice those in the first equation. Every solution to this equation will be a solution to the first equation.)
A pair of linear equations has no solution when they describe parallel lines. Parallel lines have no point of intersection.
Example
x + y = 6
x + y = 7 (No solution to this equation will also be a solution to the first equation)
_____
A single equation can have infinitely many solutions. When there are more variables than there are equations, that will generally be the case (but not always).
Example x + y = 6
A single equation may have no solutions. Usually a function is involved that has a domain or range that is not infinite.
Example x2 = -1 (The set of values produced by squaring real numbers is always positive. There does not exist a real number x that will satisfy this equation.)
