AP DSC School Assistant Math Model Test 2

Concept:For a pair of linear equations in two variables, $a_{1}x + b_{1}y = c_{1}$ and $a_{2}x + b_{2}y = c_{2}$, the system has infinitely many solutions (consistent and dependent) if $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$.Explanation:Step 1: Write the given equations in standard form:$3x - 4y = 5$ and $12x - 16y = 20$.Step 2: Compare the ratios of coefficients and constants:$$\frac{a_{1}}{a_{2}} = \frac{3}{12} = \frac{1}{4},\qquad \frac{b_{1}}{b_{2}} = \frac{-4}{-16} = \frac{1}{4},\qquad \frac{c_{1}}{c_{2}} = \frac{5}{20} = \frac{1}{4}.$$Step 3: Since all three ratios are equal ($\frac{1}{4} = \frac{1}{4} = \frac{1}{4}$), the two equations are actually the same line (the second equation is just the first multiplied by 4). Therefore, every point on the line is a common solution.Step 4: This means the system has infinitely many solutions, which is "more than two common solutions".Answer:Option D: more than two common solutions.