consider the equation y=mx+c. suppose the slope m is given as a function of x, specifically m=2x-3. derive teh equation of the line for y when c=5

Given the equation of a line \( y = mx + c \) and the slope \( m \) as a function of \( x \), specifically \( m = 2x - 3 \), we can substitute the expression for \( m \) into the original equation. Additionally, we are given that \( c = 5 \).

### Step-by-Step Derivation:

1. **Start with the general equation of the line:**
   \[
   y = mx + c
   \]

2. **Substitute the expression for \( m \) (which is given as \( m = 2x - 3 \)):**
   \[
   y = (2x - 3)x + c
   \]

3. **Expand the expression:**
   \[
   y = 2x^2 - 3x + c
   \]

4. **Substitute the value of \( c \) (which is given as \( c = 5 \)):**
   \[
   y = 2x^2 - 3x + 5
   \]

So, the equation of the line is:

\[
y = 2x^2 - 3x + 5
\]

This equation represents a parabolic curve rather than a straight line, due to the quadratic term \( 2x^2 \).

Would you like further details on this derivation or any other related questions? Here are 5 questions that might interest you:

1. How does the quadratic term affect the shape of the curve?
2. What would the graph of this equation look like?
3. How does changing the value of \( c \) affect the graph?
4. Can you find the vertex of this parabola?
5. How does the line's slope vary with \( x \)?

**Tip:** Understanding how the slope \( m \) changes with \( x \) is crucial in determining the behavior of the curve and whether it represents a straight line or a more complex shape like a parabola.

Consider the equation y = mx + c. Suppose the slope m is given as a function of x, specifically m = 2x - 3. Derive the equation of the line for y when c = 5.

Explore how to derive the equation of a line where the slope is a function of x. In this problem, we start with y = mx + c and m = 2x - 3 to obtain a quadratic equation y = 2x^2 - 3x + 5. This problem involves key algebraic concepts, including linear and quadratic equations, suitable for Grades 8-10.

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