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To solve the problem, we are asked to find the point at which two lines intersect, specifically \( g(x) = -\frac{1}{3}x + \frac{47}{18} \) and \( h(x) = -\frac{6}{5}x + \frac{81}{10} \).

### Step 1: Set the equations equal to each other

Since the two lines intersect at the same point, their \( y \)-coordinates will be equal at that point. So we set \( g(x) \) equal to \( h(x) \):

\[
-\frac{1}{3}x + \frac{47}{18} = -\frac{6}{5}x + \frac{81}{10}
\]

### Step 2: Eliminate fractions by multiplying through by a common denominator

The denominators in the equation are 3, 18, 5, and 10. The least common denominator (LCD) of 3, 18, 5, and 10 is 90. Multiply both sides of the equation by 90 to eliminate the fractions:

\[
90 \left( -\frac{1}{3}x + \frac{47}{18} \right) = 90 \left( -\frac{6}{5}x + \frac{81}{10} \right)
\]

This simplifies to:

\[
-30x + 235 = -108x + 729
\]

### Step 3: Solve for \( x \)

Now, solve the equation for \( x \):

\[
-30x + 235 = -108x + 729
\]

Add \( 108x \) to both sides:

\[
78x + 235 = 729
\]

Subtract 235 from both sides:

\[
78x = 494
\]

Now, divide by 78:

\[
x = \frac{494}{78} = \frac{247}{39}
\]

### Step 4: Substitute \( x \) back to find \( y \)

Now that we have \( x = \frac{247}{39} \), substitute this value into one of the original equations (let's use \( g(x) \)) to find \( y \):

\[
g\left( \frac{247}{39} \right) = -\frac{1}{3} \times \frac{247}{39} + \frac{47}{18}
\]

First, simplify \( -\frac{1}{3} \times \frac{247}{39} \):

\[
-\frac{247}{117}
\]

Now, we need to add \( \frac{47}{18} \) to \( -\frac{247}{117} \). First, get a common denominator (LCD of 117 and 18 is 234):

\[
-\frac{247}{117} = -\frac{494}{234}, \quad \frac{47}{18} = \frac{611}{234}
\]

Now, add them:

\[
-\frac{494}{234} + \frac{611}{234} = \frac{117}{234} = \frac{1}{2}
\]

### Final Answer:

The intersection point is \( \left( \frac{247}{39}, \frac{1}{2} \right) \).

Would you like further details, or have any questions?

Here are 5 related questions to explore further:
1. What happens if two lines are parallel? Do they intersect?
2. How can you graphically verify the intersection point?
3. What is the geometrical meaning of the slope in linear equations?
4. How do you determine if two lines are perpendicular using their slopes?
5. Can you find the intersection point of non-linear curves similarly?

**Tip:** When solving systems of linear equations, using elimination or substitution methods can often simplify the process.

Use algebra to find the point at which the line g(x) = -1/3x + 47/18 intersects the line h(x) = -6/5x + 81/10. Write the values of x and y as reduced fractions or integers.

This guide walks through solving for the intersection of two linear equations with fractional coefficients: g(x) = -1/3x + 47/18 and h(x) = -6/5x + 81/10. Suitable for students in Grades 8-10.

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