To solve this problem, let's use the intercept form of the equation of a line. The intercept form is given by:

$\frac{x}{a} + \frac{y}{b} = 1$

where $a$ is the x-intercept and $b$ is the y-intercept of the line. Given the information:

1. The line passes through the point $(6, 7)$.
2. The sum of the intercepts on the axes is 12, so $a + b = 12$.

Let's substitute the point $(6, 7)$ into the intercept form:

$\frac{6}{a} + \frac{7}{b} = 1$

Now, we also know $a + b = 12$. We can solve these two equations simultaneously.

Step-by-Step Solution:

1. Equation for the sum of intercepts:

$a + b = 12 \implies b = 12 - a$

2. Substitute $b$ in the intercept form:

$\frac{6}{a} + \frac{7}{12 - a} = 1$

3. Find a common denominator and solve for $a$:

$\frac{6(12 - a) + 7a}{a(12 - a)} = 1$

$6(12 - a) + 7a = a(12 - a)$

$72 - 6a + 7a = 12a - a^2$

$a^2 - 11a + 72 = 0$

4. Solve the quadratic equation:

The quadratic equation $a^2 - 11a + 72 = 0$ can be factored as:

$(a - 8)(a - 9) = 0$

Thus, $a = 8$ or $a = 9$.

5. Find corresponding $b$ values:

If $a = 8$, then $b = 12 - 8 = 4$.
If $a = 9$, then $b = 12 - 9 = 3$.

Equations of the Lines:

For $a = 8$ and $b = 4$:

$\frac{x}{8} + \frac{y}{4} = 1 \implies 4x + 8y = 32 \implies 4x + 3y = 24 \quad (\text{dividing by 2})$

For $a = 9$ and $b = 3$:

$\frac{x}{9} + \frac{y}{3} = 1 \implies 3x + 9y = 27 \implies x + 3y = 9$

Compare with the Options:

• Option A: $x + y = 7$ — Not correct.
• Option B: $4x + 3y = 24$ — Correct.
• Option C: $3x - 4y = 12$ — Not correct.
• Option D: None — Not correct.

The correct answer is Option B: $4x + 3y = 24$.

Would you like more details or have any questions?

Related Questions:

1. How do you determine the intercept form of a line?
2. Can a line have negative intercepts, and how does it affect its equation?
3. How do you find the equation of a line given two points instead of intercepts?
4. What is the general form of a linear equation, and how can you convert between different forms?
5. What methods can be used to solve a quadratic equation like $a^2 - 11a + 72 = 0$?

Tip:

When dealing with linear equations and intercepts, remember that intercepts represent where the line crosses the axes, and can help visualize the line's slope and position in the coordinate plane.