Find the equation of the line parallel to y equals 1 half x plus 2 that passes through the point (4, 5) in slope-intercept form.

The equation of the line given is in slope-intercept form: 

\[
y = \frac{1}{2}x + 2
\]

From this, we can identify that the slope \(m\) of the line is \( \frac{1}{2} \).

For any line parallel to this one, the slope will remain the same. Therefore, the slope of the new line that we need to find will also be \( \frac{1}{2} \).

Now, we have the slope of the new line, and we are also given the point through which the new line passes: \( (4, 5) \).

### Step 1: Use the point-slope form of the line equation
The point-slope form of the equation of a line is:

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

where \(m\) is the slope, and \((x_1, y_1)\) is the point on the line. Substituting \( m = \frac{1}{2} \), \( x_1 = 4 \), and \( y_1 = 5 \):

\[
y - 5 = \frac{1}{2}(x - 4)
\]

### Step 2: Simplify the equation to slope-intercept form
Now, simplify the equation:

\[
y - 5 = \frac{1}{2}(x - 4)
\]
\[
y - 5 = \frac{1}{2}x - 2
\]
\[
y = \frac{1}{2}x - 2 + 5
\]
\[
y = \frac{1}{2}x + 3
\]

Thus, the equation of the line parallel to \(y = \frac{1}{2}x + 2\) that passes through the point \( (4, 5) \) is:

\[
y = \frac{1}{2}x + 3
\]

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Would you like further details on this, or do you have any other questions?

### Related questions:
1. How can I find the slope of a line given two points on it?
2. What is the general form of the equation of a line?
3. How can I graph a line using its equation?
4. What is the point-slope form of a linear equation, and how is it derived?
5. How does the slope of parallel lines compare to that of perpendicular lines?

**Tip:** When writing the equation of a line in slope-intercept form, remember to identify the slope and the y-intercept clearly to avoid mistakes!

Learn how to find the equation of a line parallel to y = 1/2x + 2 that passes through the point (4, 5). This step-by-step solution uses the point-slope form of the line equation and explains the concept of parallel lines having equal slopes. Suitable for students in Grades 8-10.

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