Do the three lines:
x - 4y= 1
2x - y = -3
-x - 3y = 4  
have a common point of intersection?
Explain.

To determine whether the three lines have a common point of intersection, we need to check if their system of equations has a single solution that satisfies all three equations. The system of equations is:

\[
1) \ x - 4y = 1
\]
\[
2) \ 2x - y = -3
\]
\[
3) \ -x - 3y = 4
\]

This is a system of three linear equations in two variables. We'll solve it step by step.

### Step 1: Solve two of the equations
We will first solve equations (1) and (2) to find \(x\) and \(y\).

#### Solve equations (1) and (2):

1. From equation (1): 
   \[
   x = 1 + 4y \quad \text{(rearrange for \(x\))}
   \]

2. Substitute this expression for \(x\) into equation (2):
   \[
   2(1 + 4y) - y = -3
   \]
   Expanding this:
   \[
   2 + 8y - y = -3
   \]
   \[
   7y = -5
   \]
   \[
   y = -\frac{5}{7}
   \]

3. Substitute \(y = -\frac{5}{7}\) into \(x = 1 + 4y\):
   \[
   x = 1 + 4\left(-\frac{5}{7}\right)
   \]
   \[
   x = 1 - \frac{20}{7} = \frac{7}{7} - \frac{20}{7} = -\frac{13}{7}
   \]

So, from the first two equations, we get \(x = -\frac{13}{7}\) and \(y = -\frac{5}{7}\).

### Step 2: Check the solution in the third equation
Now we need to verify if these values of \(x\) and \(y\) also satisfy the third equation \(-x - 3y = 4\).

Substitute \(x = -\frac{13}{7}\) and \(y = -\frac{5}{7}\) into the third equation:
\[
-(-\frac{13}{7}) - 3\left(-\frac{5}{7}\right) = 4
\]
Simplifying:
\[
\frac{13}{7} + \frac{15}{7} = 4
\]
\[
\frac{28}{7} = 4
\]
\[
4 = 4
\]

Since this equation holds true, the values \(x = -\frac{13}{7}\) and \(y = -\frac{5}{7}\) satisfy all three equations.

### Conclusion:
Yes, the three lines have a common point of intersection at \( \left(-\frac{13}{7}, -\frac{5}{7}\right) \).

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you solve a system of linear equations graphically?
2. Can a system of three linear equations have no solution?
3. What is the geometric interpretation of having infinitely many solutions?
4. How can you determine if a system is consistent or inconsistent?
5. How does the substitution method compare to the elimination method for solving systems?

**Tip:** Always verify your solution in every original equation to ensure it satisfies the entire system.

Do the three lines: x - 4y = 1, 2x - y = -3, -x - 3y = 4 have a common point of intersection? Explain.

Explore how to determine whether three lines represented by x - 4y = 1, 2x - y = -3, and -x - 3y = 4 have a common point of intersection. This problem demonstrates the use of substitution in solving systems of linear equations, ensuring the equations are consistent and finding the intersection point. Suitable for high school students in Grades 9-12.

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