4.4 Q-3

Question Statement

Find an equation of the line that passes through the intercepts of the following two lines:

• $16x - 10y - 33 = 0$
• $12x + 14y + 29 = 0$

and also through the intersection of the lines:

• $x - y + 4 = 0$
• $x - 7y + 2 = 0$

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Background and Explanation

To solve this, we need to:

1. Find the point of intersection of the first pair of lines $16x - 10y - 33 = 0$ and $12x + 14y + 29 = 0$.
2. Find the point of intersection of the second pair of lines $x - y + 4 = 0$ and $x - 7y + 2 = 0$.
3. Using these points, we will then find the equation of the line that passes through both the intercepts and the intersection points.

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Solution

Step 1: Find the intersection of the lines $16x - 10y - 33 = 0$ and $12x + 14y + 29 = 0$

We have the following system of equations:

1. $16x - 10y - 33 = 0$
2. $12x + 14y + 29 = 0$

To solve this system, we’ll use the method of elimination. First, we multiply both equations to eliminate one variable. After simplifying the system, we obtain:

$$\frac{x}{172} = \frac{-y}{860} = \frac{1}{344}$$

From this, we find the values for $x$ and $y$:

$$x = \frac{172}{344} = \frac{1}{2}, \quad y = \frac{-860}{344} = -\frac{5}{2}$$

Thus, the point of intersection is:

$$\left( \frac{1}{2}, -\frac{5}{2} \right)$$

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Step 2: Find the equation of the line passing through the point $\left( \frac{1}{2}, -\frac{5}{2} \right)$ and the intersection of $x - y + 4 = 0$ and $x - 7y + 2 = 0$

Now we need to find the intersection of the second pair of lines:

1. $x - y + 4 = 0$
2. $x - 7y + 2 = 0$

We combine these equations as follows:

$$(x - y + 4) + K(x - 7y + 2) = 0 \tag{1}$$

Substituting the point $\left( \frac{1}{2}, -\frac{5}{2} \right)$ into this equation:

$$\left[ \frac{1}{2} - \frac{5}{2} + 4 \right] + K \left[ \frac{1}{2} - 7 - \frac{5}{2} + 2 \right] = 0$$

Simplifying:

$$2 - 15K = 0 \quad \Rightarrow \quad K = \frac{2}{15}$$

Now, substitute $K = \frac{2}{15}$ into equation (1):

$$(x - y + 4) + \frac{2}{15}(x - 7y + 2) = 0$$

Multiplying through by 15 to eliminate the fraction:

$$15(x - y + 4) + 2(x - 7y + 2) = 0$$

Simplifying:

$$15x + 15y + 60 - 2x + 14y + 6 = 0$$

Combine like terms:

$$13x + 29y + 66 = 0$$

Thus, the equation of the required line is:

$$\boxed{13x + 29y + 66 = 0}$$

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Key Formulas or Methods Used

• Point of Intersection: We solve two linear equations simultaneously to find the point of intersection.
• Equation of Line Through Two Points: We use the intersection points and combine the equations to find the required line.

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Summary of Steps

1. Find the point of intersection of $16x - 10y - 33 = 0$ and $12x + 14y + 29 = 0$.
2. Find the point of intersection of $x - y + 4 = 0$ and $x - 7y + 2 = 0$.
3. Use the point of intersection to find the equation of the line passing through both the intercepts and the intersection points.
4. Final answer: $13x + 29y + 66 = 0$.