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options are A.  X+Y = 7       B.   4X+3Y =24                C.     3X-4Y =12             D. NONE

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.

Explore how to solve linear equations using intercept forms and simultaneous equations. This problem involves finding intercepts and determining equations of lines passing through given points. Suitable for high school students in Grades 9-12.

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