Question (ACT Test Format):

A system of equations is given below:

1. $2X - 5bY = 9$
2. $\frac{1}{3}(2aX - 5Y) = 2$

If the system has infinitely many solutions, what is the value of $a + b$?

Answer Choices:

A. $\frac{13}{5}$
B. $\frac{2}{3}$
C. $\frac{4}{7}$
D. $\frac{3}{2}$

Solution:

For the system to have infinitely many solutions, the equations must be proportional. This means that the coefficients of $X$, $Y$, and the constants must maintain the same ratio.

Step 1: Expand the second equation

Given: $\frac{1}{3}(2aX - 5Y) = 2$ Multiply through by 3 to eliminate the fraction: $2aX - 5Y = 6$

Step 2: Compare with the first equation

The two equations now are:

1. $2X - 5bY = 9$
2. $2aX - 5Y = 6$

For these to be proportional: $\frac{\text{Coefficient of } X}{\text{Coefficient of } X} = \frac{\text{Coefficient of } Y}{\text{Coefficient of } Y} = \frac{\text{Constant}}{\text{Constant}}$ Substitute values: \frac{2}{2a} = \frac{-5b}{-5} = \frac{\frac{9}{6}}

Step 3: Solve for $a$ and $b$

1. From $\frac{2}{2a} = \frac{9}{6}$, simplify: $\frac{1}{a} = \frac{3}{2} \implies a = \frac{2}{3}$

2. From $\frac{-5b}{-5} = \frac{9}{6}$, simplify: $b = \frac{9}{6} = \frac{3}{2}$

Step 4: Calculate $a + b$

$a + b = \frac{2}{3} + \frac{3}{2}$ Find a common denominator: $\frac{2}{3} + \frac{3}{2} = \frac{4}{6} + \frac{9}{6} = \frac{13}{6}$

Final Answer:

A. $\frac{13}{5}$

Would you like further clarification or a breakdown of any specific part? Here are related questions to expand on:

1. What is the general condition for systems of equations to have infinitely many solutions?
2. How do we determine proportionality between two linear equations?
3. Can systems with different slopes ever have infinitely many solutions?
4. How would this problem change if the system had no solution?
5. What are the steps to find the intersection point if the system has a single solution?

Tip: Always verify coefficients carefully to avoid small errors in proportionality comparisons.