Consistent System of Equations:

A consistent system of equations means that they intersect. There are two kinds of situations when a consistent system is a result - when the equations are intersecting or when the equations are the same.

Let {eq}q {/eq} stand for the number of quarters and {eq}d {/eq} stand for the number of dimes. The total of their value is $6.65 and as an equation would be: {eq}\$0.25q + \$0.10d = \$6.65 \rm \ Eq. 1 {/eq}

The number of quarters is 7 less than twice the number of dimes, as an equation it would be:

{eq}q = 2d -7 \rm \ Eq.2 {/eq}

Substituting Eq. 2 in Eq. 1:

{eq}\begin{align} \$0.25q + \$0.10d &= \$6.65 \\ \$0.25(2d -7) + \$0.10d &= \$6.65 \\ \$0.50d - \$1.75 + \$0.10d &= \$6.65 \\ \$0.60d &= \$6.65 + \$1.75 \\ \$0.60d &= \$8.40 \\ \dfrac{\$0.60d}{\$0.60} &= \dfrac{\$8.40}{\\$0.60} \\ d &= 14 \end{align} {/eq}

There are 14 dimes and substituting this in Eq. 2:

{eq}q = 2d -7 \\ q = 2(14) -7 \\ q = 28 - 7 \\ q = 21 {/eq}

There are 14 dimes and 21 quarters.