Question 209910
Let a = the larger number and b = the smaller number.
{{{a*b = 1}}} and...
{{{a-b = 3/2}}} Rewrite this as {{{a = b+3/2}}} and substitute into the first equation.
{{{(b+3/2)*b = 1}}} Simplify.
{{{b^2+(3/2)b = 1}}} Subtract 1 from both sides.
{{{b^2+(3/2)b-1 = 0}}} Multiply through by 2 to clear the fraction.
{{{2b^2+3b-2 = 0}}} Factor this trinomial.
{{{(2b-1)(b+2) = 0}}} Apply the zero product rule.
{{{2b-1 = 0}}} or {{{b+2 = 0}}} so that...
{{{2b = 1}}} or {{{b = -2}}} Discard the negative solution as the numbers are to be positive.
{{{2b = 1}}} Divide both sides by 2.
{{{b = 1/2}}} This is the smaller number.
{{{a = b+3/2}}} Substitute b = 1/2.
{{{a = 1/2 + 3/2}}}
{{{highlight(b = 2)}}} This is the larger number.
Check:
{{{a*b = (2)*(1/2)}}}
{{{a*b = 1}}} and...
{{{a-b = 2-1/2}}}
{{{a-b = 3/2}}}