Show that the lines `(x-4)/(1)=(y+3)/(-4)=(z+1)/7 and (x-1)/(2) =(y+1)/(-3)=(z+10)/(8)` intersect. Find the coordinates of their point of intersection.

Answer

To show that the lines

\[ \frac{x-4}{1} = \frac{y+3}{-4} = \frac{z+1}{7} \]

and

\[ \frac{x-1}{2} = \frac{y+1}{-3} = \frac{z+10}{8} \]

intersect, we will find the coordinates of their point of intersection step by step.

### Step 1: Parametrize the Lines

**Line 1** can be expressed in parametric form as:
\[ \begin{align*} x &= 4 + \lambda \\ y &= -3 - 4\lambda \\ z &= -1 + 7\lambda \end{align*} \]

**Line 2** can be expressed in parametric form as:
\[ \begin{align*} x &= 1 + 2\mu \\ y &= -1 - 3\mu \\ z &= -10 + 8\mu \end{align*} \]

### Step 2: Set the Coordinates Equal

For the lines to intersect, the coordinates must be equal at some values of \(\lambda\) and \(\mu\). Thus, we set the equations equal to each other:

1. From the x-coordinates:
\[ 4 + \lambda = 1 + 2\mu \quad \text{(Equation 1)} \]

2. From the y-coordinates:
\[ -3 - 4\lambda = -1 - 3\mu \quad \text{(Equation 2)} \]

3. From the z-coordinates:
\[ -1 + 7\lambda = -10 + 8\mu \quad \text{(Equation 3)} \]

### Step 3: Solve the Equations

**From Equation 1:**
\[ \lambda - 2\mu = -3 \quad \text{(Equation 4)} \]

**From Equation 2:**
\[ 3\mu - 4\lambda = 2 \quad \text{(Equation 5)} \]

### Step 4: Solve Equations 4 and 5

We can solve Equations 4 and 5 simultaneously.

From Equation 4, we can express \(\lambda\):
\[ \lambda = 2\mu - 3 \]

Substituting \(\lambda\) into Equation 5:
\[ 3\mu - 4(2\mu - 3) = 2 \]
\[ 3\mu - 8\mu + 12 = 2 \]
\[ -5\mu + 12 = 2 \]
\[ -5\mu = -10 \implies \mu = 2 \]

### Step 5: Substitute \(\mu\) Back to Find \(\lambda\)

Substituting \(\mu = 2\) back into Equation 4:
\[ \lambda = 2(2) - 3 = 4 - 3 = 1 \]

### Step 6: Verify with the Third Coordinate

Now we will check if these values satisfy Equation 3:
\[ -1 + 7(1) = -10 + 8(2) \]
\[ -1 + 7 = -10 + 16 \]
\[ 6 = 6 \quad \text{(True)} \]

### Step 7: Find the Point of Intersection

Now that we have \(\lambda = 1\) and \(\mu = 2\), we can find the coordinates of the intersection point using either line. We'll use Line 1:

\[ \begin{align*} x &= 4 + 1 = 5 \\ y &= -3 - 4(1) = -7 \\ z &= -1 + 7(1) = 6 \end{align*} \]

Thus, the point of intersection is \((5, -7, 6)\).

### Conclusion

The lines intersect at the point \((5, -7, 6)\).

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