For a system of linear equations to have infinite solutions, the two equations must be proportional to each other. This means that one equation is a scalar multiple of the other.

We are given the following system of equations:

1. $2x + 3y = 7$
2. $2ax + (a + b)y = 28$

For infinite solutions, the ratios of the corresponding coefficients of $x$, $y$, and the constants must be equal. That is:

$\frac{2a}{2} = \frac{a + b}{3} = \frac{28}{7}$

Step 1: Compare the constants

From the third ratio, we can compare the constants:

$\frac{28}{7} = 4$

Thus, the scalar multiplier is 4.

Step 2: Set up the proportional equations

Using this scalar multiplier, we now compare the coefficients of $x$ and $y$:

1. $\frac{2a}{2} = 4$, which simplifies to: $a = 4$

2. $\frac{a + b}{3} = 4$, which simplifies to: $a + b = 12$

Step 3: Solve for $b$

Since $a = 4$, substitute this into the equation $a + b = 12$:

$4 + b = 12$

Solving for $b$:

$b = 12 - 4 = 8$

Final Answer:

The values of $a$ and $b$ are:

$a = 4, \quad b = 8$

Would you like more details or have any questions?

Here are 5 related questions:

1. How do we determine if a system of equations has no solutions?
2. What conditions lead to a system having exactly one solution?
3. Can you explain the geometric interpretation of infinite solutions for a system of linear equations?
4. How do you solve systems of linear equations using substitution?
5. How does elimination work in solving systems of linear equations?

Tip: Inconsistent systems have parallel lines that never intersect, indicating no solution.