If the system of linear equations 2x + 3y = 7 and 2ax + (a + b)y = 28 has infinite number of 
solutions, then find the values of a and b.

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.

If the system of linear equations 2x + 3y = 7 and 2ax + (a + b)y = 28 has infinite number of solutions, then find the values of a and b.

Explore how to find the values of a and b for the system of linear equations 2x + 3y = 7 and 2ax + (a + b)y = 28, where the system has infinite solutions. This solution includes the concept of proportionality, key formulas, and step-by-step explanations. Suitable for students in Grades 9-11.

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