Find the value of "  " such that the following system of linear equations
have
i) exactly one solution
ii) No solution
iii) Infinite number of solutions and find them.
step by step solve

To solve this problem, we need to determine the conditions under which the given system of linear equations has:

1. **Exactly one solution**
2. **No solution**
3. **An infinite number of solutions**

Let’s denote the system of linear equations as follows:

\[
a_1x + b_1y = c_1
\]
\[
a_2x + b_2y = c_2
\]

We'll analyze the system based on the relationship between \(a_1\), \(b_1\), \(c_1\) and \(a_2\), \(b_2\), \(c_2\).

### Step 1: Write Conditions for Each Case

To determine the conditions for each case, we’ll use the **determinant** of the coefficients of \(x\) and \(y\). For a system of two linear equations to have:

1. **Exactly One Solution**: The determinant of the coefficient matrix should be non-zero.
2. **No Solution**: The lines represented by the equations should be parallel (same slope) but have different intercepts.
3. **Infinite Number of Solutions**: The two equations should represent the same line, meaning they are dependent.

Let’s break down each case with the specific criteria:

#### i) **Exactly One Solution**

For the system to have exactly one solution, the coefficient matrix must have a non-zero determinant:

\[
\text{Determinant} = a_1b_2 - a_2b_1 \neq 0
\]

This means the two lines are neither parallel nor identical, so they intersect at a unique point.

#### ii) **No Solution**

For the system to have no solution, the lines must be parallel but distinct. This happens if:

\[
\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}
\]

This indicates that the lines have the same slope but different intercepts.

#### iii) **Infinite Number of Solutions**

For the system to have infinitely many solutions, the two equations must be proportional, representing the same line. This requires:

\[
\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}
\]

This implies that both equations describe the same line, so there are infinitely many points of intersection (i.e., all points on the line).

### Step 2: Applying to the Given Equations

If you provide the specific values or equations for the coefficients \(a_1\), \(b_1\), \(c_1\), \(a_2\), \(b_2\), and \(c_2\), we can substitute them here to solve for the cases.

Let me know if you’d like to continue with specific values or need further clarification on any steps.

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Would you like more detailed steps or have any questions?

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#### Related Questions:
1. How do you find the determinant of a 2x2 matrix?
2. How do you identify if two lines are parallel or identical?
3. What are the properties of a consistent system of equations?
4. How does the determinant determine the number of solutions?
5. How can you graphically interpret the solutions of a system of equations?

#### Tip:
When dealing with systems of linear equations, always check the determinant first for an efficient way to assess solution types.

Find the value of "  " such that the following system of linear equations have i) exactly one solution ii) No solution iii) Infinite number of solutions and find them. Step by step solve.

Understand how to find values that determine the type of solutions in a system of linear equations, covering conditions for exactly one solution, no solution, and infinite solutions. This problem explores linear algebra concepts like determinants and system consistency, with step-by-step guidance. Suitable for high school students in Grades 10-12.

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