Determine the value of b if the slope f(x) of is 2, and f(2) =1 while f(b)=4 ?

To find the value of \( b \), we need to determine the equation of the linear function \( f(x) \) given the slope and two points on the line.

### Step 1: Use the slope-intercept form
The slope-intercept form of a linear equation is:
\[
f(x) = mx + c
\]
where \( m \) is the slope and \( c \) is the y-intercept.

### Step 2: Substitute the given slope
We are given that the slope \( m = 2 \), so the equation becomes:
\[
f(x) = 2x + c
\]

### Step 3: Use the point \( f(2) = 1 \) to find \( c \)
Substitute \( x = 2 \) and \( f(2) = 1 \) into the equation:
\[
1 = 2(2) + c
\]
\[
1 = 4 + c
\]
\[
c = 1 - 4 = -3
\]

### Step 4: Write the equation of the line
Now that we have \( c = -3 \), the equation of the line is:
\[
f(x) = 2x - 3
\]

### Step 5: Use \( f(b) = 4 \) to find \( b \)
Substitute \( f(b) = 4 \) into the equation:
\[
4 = 2b - 3
\]
\[
2b = 4 + 3
\]
\[
2b = 7
\]
\[
b = \frac{7}{2} = 3.5
\]

### Conclusion
The value of \( b \) is \( \boxed{3.5} \).

Would you like further details or have any questions?

Here are some related questions:
1. What is the general form of the equation of a line?
2. How do you find the y-intercept from two points on a line?
3. What is the significance of the slope in the equation of a line?
4. How do you solve for a variable in a linear equation?
5. Can you find \( b \) if the slope was different?

**Tip:** When dealing with linear equations, always remember that the slope remains constant for any two points on the line.

Learn how to find the value of b in a linear equation f(x) = 2x - 3, where f(2) = 1 and f(b) = 4. This problem involves using the slope-intercept form and solving for a variable in a linear equation. Suitable for students in Grades 7-9.

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