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To find the equation of the tangent line to the graph of \( y = g(x) \) at \( x = 5 \), given that \( g(5) = -3 \) and \( g'(5) = 4 \), we can use the point-slope form of a line:

\[
y - y_1 = m(x - x_1)
\]

where:
- \( (x_1, y_1) \) is the point on the curve, which is \( (5, g(5)) = (5, -3) \),
- \( m \) is the slope of the tangent line, which is \( g'(5) = 4 \).

### Step-by-Step Solution

1. **Identify the point** on the tangent line:
   \[
   (x_1, y_1) = (5, -3)
   \]

2. **Identify the slope** of the tangent line:
   \[
   m = g'(5) = 4
   \]

3. **Substitute into the point-slope form** of the equation:
   \[
   y - (-3) = 4(x - 5)
   \]

4. **Simplify the equation**:
   \[
   y + 3 = 4(x - 5)
   \]
   \[
   y + 3 = 4x - 20
   \]
   \[
   y = 4x - 23
   \]

### Final Answer
The equation of the tangent line is:
\[
y = 4x - 23
\]

Would you like more details or have any questions?

### Related Questions
1. How do you find the tangent line if \( g'(x) \) is not provided?
2. How does the tangent line equation change if we use different forms of linear equations?
3. What is the geometric interpretation of \( g'(5) = 4 \)?
4. How would the equation change if the point \( (5, -3) \) was different?
5. How do you find the normal line to \( y = g(x) \) at \( x = 5 \)?

### Tip
The slope of the tangent line at a point is always equal to the value of the derivative at that point.

Find an equation of the tangent line to the graph of y = g(x) at x = 5 if g(5) = -3 and g'(5) = 4.

Find the equation of the tangent line to y = g(x) at x = 5, given g(5) = -3 and g'(5) = 4. This calculus problem explores derivatives and the point-slope form for tangent line equations, suitable for high school students in Grades 11-12.

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