https://file.aiquickdraw.com/mathbot/file/1730083348512exse21rb.jpg

The system of linear equations given in the image is:

\[
\begin{cases}
ax - y = -1 \\
-4x + 2y = 3
\end{cases}
\]

The problem asks to find the value of \( a \) for which this system of equations has no solution.

### Step-by-Step Solution

To determine when a system of equations has no solution, we look for conditions that make the lines represented by the equations parallel but not coincident. In this case, parallel lines imply that the system is inconsistent and therefore has no solutions.

#### Step 1: Rewrite the equations in standard form

1. The first equation is already in a suitable form:
   \[
   ax - y = -1
   \]

2. Rewrite the second equation in standard form:
   \[
   -4x + 2y = 3
   \]

#### Step 2: Convert to slope-intercept form

To determine if the lines are parallel, we can convert each equation to slope-intercept form \( y = mx + b \) and compare their slopes.

1. **First Equation:** Solve for \( y \):
   \[
   y = ax + 1
   \]
   The slope of the first equation is \( m_1 = a \).

2. **Second Equation:** Solve for \( y \):
   \[
   2y = 4x + 3
   \]
   Divide by 2:
   \[
   y = 2x + \frac{3}{2}
   \]
   The slope of the second equation is \( m_2 = 2 \).

#### Step 3: Set slopes equal for parallel lines

For the lines to be parallel, their slopes must be equal:
\[
a = 2
\]

#### Step 4: Check for inconsistency

If \( a = 2 \), the system becomes:
\[
\begin{cases}
2x - y = -1 \\
-4x + 2y = 3
\end{cases}
\]

Multiplying the first equation by 2, we get:
\[
4x - 2y = -2
\]

Now we have:
\[
\begin{cases}
4x - 2y = -2 \\
-4x + 2y = 3
\end{cases}
\]

Adding these two equations gives:
\[
0 = 1
\]
which is a contradiction. Thus, the system has no solution when \( a = 2 \).

### Answer
The value of \( a \) for which the system has no solution is:
\[
\boxed{a = 2}
\]

Would you like a more detailed explanation or have any further questions?

---

Here are five additional questions to expand your understanding:

1. What does it mean for a system of linear equations to be inconsistent?
2. How can you determine if a system of linear equations has infinitely many solutions?
3. What role does the slope play in determining the relationship between two lines?
4. What happens if the two lines represented by a system have the same slope and the same intercept?
5. How can we solve a system of equations if we know it has a unique solution?

**Tip:** When checking for no solutions in a system of linear equations, remember to examine if the slopes are equal but the intercepts differ.

Тэгштгэлэйн систем шийдгүи байх \( a \)-г ол.

Discover how to find the value of 'a' for which the given system of linear equations has no solution. This problem explores algebraic concepts like slopes, parallel lines, and system inconsistency, making it ideal for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation