SAT Mathematics Practice Test 2

In order for a system of two linear equations to have infinitely many solutions, the two equations must be equivalent. Thus,the equation $\mathit{a}\mathit{x}+\mathit{b}\mathit{y}=12$ must be equivalent to the equation $2\mathit{x}+8\mathit{y}=60$ Multiplying each side of $\mathit{a}\mathit{x}+\mathit{b}\mathit{y}=12$ by 5 gives $5\mathit{a}\mathit{x}+5\mathit{b}\mathit{y}=60$ which must be equivalent to $2\mathit{x}+8\mathit{y}=60$ .Since the right-hand sides of $5\mathit{a}\mathit{x}+5\mathit{b}\mathit{y}=60$ and $2\mathit{x}+8\mathit{y}=60$ are the same, equating coefficients gives $5\mathit{a}=2,$ or $\mathit{a}=\frac{2}{5}$ and $5\mathit{b}=8$ ,or $\mathit{b}=\frac{8}{5}$.Therefore,the value of $\left(\frac{\mathit{a}}{\mathit{b}}\right)=\left(\frac{2}{5}\right)÷\left(\frac{8}{5}\right)$,which is equal to $\frac{1}{4}$.Either the fraction $\frac{1}{4}$ or its equivalent decimal,25 may be gridded as the correct answer.
Alternatively, since $\mathit{a}\mathit{x}+\mathit{b}\mathit{y}=12$ is equivalent to $2\mathit{x}+8\mathit{y}=60$,the equation $\mathit{a}\mathit{x}+\mathit{b}\mathit{y}=12$ is equal to $2\mathit{x}+8\mathit{y}=60$ multiplied on each side by the same constant. Since multiplying $2\mathit{x}=8\mathit{y}=60$ by a constant does not change the ratio of the coefficient of $\mathit{x}$ to the coefficient of y,it follows that $\frac{\mathit{a}}{\mathit{b}}=\frac{2}{8}=\frac{1}{4}$