Question

question 4
(04.02 mc)
determine which system will produce infinitely many solutions.
○ 2x + 5y = 24
2x + 5y = 42
○ 3x - 2y = 15
6x + 5y = 11
○ 4x - 3y = 9
-8x + 6y = -18
○ 5x - 3y = 16
-2x + 3y = -7

Explanation:

Step1: Recall the condition for infinite solutions

A system of linear equations \(a_1x + b_1y = c_1\) and \(a_2x + b_2y = c_2\) has infinitely many solutions if \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\) (the two equations are scalar multiples of each other).

Step2: Check Option 1

For \(2x + 5y = 24\) and \(2x + 5y = 42\), \(\frac{2}{2}=\frac{5}{5}=1\), but \(\frac{24}{42}=\frac{4}{7}
eq1\). So no infinite solutions.

Step3: Check Option 2

For \(3x - 2y = 15\) and \(6x + 5y = 11\), \(\frac{3}{6}=\frac{1}{2}\), \(\frac{-2}{5}
eq\frac{1}{2}\). So no infinite solutions.

Step4: Check Option 3

For \(4x - 3y = 9\) and \(-8x + 6y = -18\), \(\frac{4}{-8}=-\frac{1}{2}\), \(\frac{-3}{6}=-\frac{1}{2}\), \(\frac{9}{-18}=-\frac{1}{2}\). All ratios are equal.

Step5: Check Option 4 (for completeness)

For \(5x - 3y = 16\) and \(-2x + 3y = -7\), \(\frac{5}{-2}
eq\frac{-3}{3}\). So no infinite solutions.

Answer:

The system \(4x - 3y = 9\) and \(-8x + 6y = -18\) (the third option) will produce infinitely many solutions.