Find the equation of the straight line passing through the point (10, - 7) and making intercepts on the coordinate axes whose sum is 12.

Answer

To find the equation of the straight line passing through the point (10, -7) and making intercepts on the coordinate axes whose sum is 12, we can follow these steps:

### Step 1: Understand the intercept form of the equation of a line
The equation of a line in intercept form is given by:
\[ \frac{x}{a} + \frac{y}{b} = 1 \]
where \(a\) is the x-intercept and \(b\) is the y-intercept.

### Step 2: Set up the relationship between intercepts
According to the problem, the sum of the intercepts is given as:
\[ a + b = 12 \]
From this, we can express \(b\) in terms of \(a\):
\[ b = 12 - a \]

### Step 3: Substitute the intercepts into the line equation
Substituting \(b\) into the intercept form gives:
\[ \frac{x}{a} + \frac{y}{12 - a} = 1 \]

### Step 4: Substitute the point (10, -7) into the equation
Since the line passes through the point (10, -7), we substitute \(x = 10\) and \(y = -7\) into the equation:
\[ \frac{10}{a} + \frac{-7}{12 - a} = 1 \]

### Step 5: Clear the fractions
To eliminate the fractions, we can multiply through by \(a(12 - a)\):
\[ 10(12 - a) - 7a = a(12 - a) \]

### Step 6: Expand and rearrange the equation
Expanding both sides gives:
\[ 120 - 10a - 7a = 12a - a^2 \]
Combining like terms results in:
\[ 120 - 17a = 12a - a^2 \]
Rearranging leads to:
\[ a^2 - 29a + 120 = 0 \]

### Step 7: Factor the quadratic equation
Now, we need to factor the quadratic equation:
\[ a^2 - 29a + 120 = 0 \]
This can be factored as:
\[ (a - 24)(a - 5) = 0 \]
Thus, the solutions for \(a\) are:
\[ a = 24 \quad \text{or} \quad a = 5 \]

### Step 8: Find corresponding values of \(b\)
Using \(b = 12 - a\):
- If \(a = 24\), then \(b = 12 - 24 = -12\).
- If \(a = 5\), then \(b = 12 - 5 = 7\).

### Step 9: Write the equations of the lines
Now we can write the equations of the lines using the intercepts:
1. For \(a = 24\) and \(b = -12\):
\[ \frac{x}{24} + \frac{y}{-12} = 1 \implies \frac{x}{24} - \frac{y}{12} = 1 \]
Multiplying through by 24 gives:
\[ x - 2y = 24 \]

2. For \(a = 5\) and \(b = 7\):
\[ \frac{x}{5} + \frac{y}{7} = 1 \]
Multiplying through by 35 gives:
\[ 7x + 5y = 35 \]

### Final Answer
The equations of the straight lines are:
1. \(x - 2y = 24\)
2. \(7x + 5y = 35\)