Solving Systems for Infinitely Many Solutions

A system of two linear equations, represented as:

$a_1x + b_1y = c_1$

$a_2x + b_2y = c_2$

has infinitely many solutions if the coefficients and constants are proportional. The condition is:

$ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} $

Applying the Condition to Given Equations

The given system is:

1. $3x - 2y = 8$

2. $2ax + (a-b)y = 48$

From equation (1), we have $a_1 = 3$, $b_1 = -2$, $c_1 = 8$.

From equation (2), we have $a_2 = 2a$, $b_2 = a-b$, $c_2 = 48$.

Calculating the Relationship Between Coefficients

Using the condition for infinitely many solutions:

$ \frac{3}{2a} = \frac{-2}{a-b} = \frac{8}{48} $

First, simplify the ratio involving constants:

$ \frac{8}{48} = \frac{1}{6} $

Now, equate the other ratios to $\frac{1}{6}$:

1. Equating the x-coefficients ratio: $ \frac{3}{2a} = \frac{1}{6} $ Cross-multiplying gives: $ 3 \times 6 = 1 \times 2a $ $ 18 = 2a $ $ a = 9 $
2. Equating the y-coefficients ratio: $ \frac{-2}{a-b} = \frac{1}{6} $ Cross-multiplying gives: $ -2 \times 6 = 1 \times (a-b) $ $ -12 = a-b $

Substitute the value of $a=9$ into the equation $-12 = a-b$:

$ -12 = 9 - b $

Rearrange to solve for $b$:

$ b = 9 + 12 $

$ b = 21 $

Verifying the Solution Option

We need to find the relationship between $a$ and $b$. We found $a=9$ and $b=21$. Let's check the given options:

• Option 1: $a = \frac{b}{3} + 2$
Substitute $a=9$ and $b=21$: $ 9 \stackrel{?}{=} \frac{21}{3} + 2 $ $ 9 \stackrel{?}{=} 7 + 2 $ $ 9 = 9 $ This option is correct.

The correct relation derived from the condition of infinitely many solutions is $a = \frac{b}{3} + 2$.