document.write( "Question 182416: What similaries and differences are there between functions and linear equations? Are all linear equations functions ? \n" ); document.write( "

Algebra.Com's Answer #136904 by solver91311(24713) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "A linear equation is a symbolic expression of a set of ordered pairs that will graph to a straight line. A function is a symbolic expression of a set of ordered pairs that will graph to any sort of line with the proviso that there is exactly one value of the function for any value of the independent variable in the domain set.\r
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\n" ); document.write( "\n" ); document.write( "So, linear equations, when expressed as

are functions that can be expressed as

. However, not all linear equations can be expressed as

. There is a set of linear equations with infinite cardinality for which the slope, m, is undefined, namely vertical lines represented by an equation of the form

. Notice that for

, there is only one possible input value, a, and an infinite number of possible output values because the point (a,y) is a point on the line regardless of the value of y. That violates the 'one input value yields exactly one output value' provisio of the definition of a function. Hence, all linear equations except those linear equations that represent graphs of vertical lines are functions, and those linear equations that represent graphs of vertical lines are not functions.\r
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