To find the equation of a line that passes through two given points, we'll follow these steps:
### Step 1: Identify the given points
The points given are [tex]\((4, 3)\)[/tex] and [tex]\((2, 2)\)[/tex]. We'll denote these points as:
- [tex]\((x_1, y_1) = (4, 3)\)[/tex]
- [tex]\((x_2, y_2) = (2, 2)\)[/tex]
### Step 2: Calculate the slope (m)
The slope of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is calculated using the formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the given points into the formula:
[tex]\[ m = \frac{2 - 3}{2 - 4} \][/tex]
[tex]\[ m = \frac{-1}{-2} \][/tex]
[tex]\[ m = \frac{1}{2} \][/tex]
### Step 3: Find the y-intercept (b)
To find the y-intercept ([tex]\(b\)[/tex]), we use the point-slope form of the equation:
[tex]\[ y = mx + b \][/tex]
We can use either of the given points. Let's use [tex]\((4, 3)\)[/tex] for this calculation.
Substitute [tex]\(m\)[/tex], [tex]\(x\)[/tex], and [tex]\(y\)[/tex] into the equation:
[tex]\[ 3 = \frac{1}{2} \cdot 4 + b \][/tex]
[tex]\[ 3 = 2 + b \][/tex]
[tex]\[ b = 3 - 2 \][/tex]
[tex]\[ b = 1 \][/tex]
### Step 4: Write the equation in slope-intercept form
Now that we have the slope ([tex]\(m = \frac{1}{2}\)[/tex]) and the y-intercept ([tex]\(b = 1\)[/tex]), we can write the equation of the line in the form [tex]\(y = mx + b\)[/tex]:
[tex]\[ y = \frac{1}{2}x + 1 \][/tex]
### Conclusion
The correct equation of the line that passes through the points [tex]\((4, 3)\)[/tex] and [tex]\((2, 2)\)[/tex] is:
[tex]\[ y = \frac{1}{2}x + 1 \][/tex]
Among the given options, the correct one is:
[tex]\[ y = \frac{1}{2}x + 1 \][/tex]
Therefore, the answer is:
[tex]\[ \boxed{y = \frac{1}{2} x + 1} \][/tex]
