Sure, let's determine the equation of the line that passes through the points [tex]\((4, 3)\)[/tex] and [tex]\((2, 2)\)[/tex]. We are given four candidate equations:
1. [tex]\(y = \frac{1}{2} x + 2\)[/tex]
2. [tex]\(y = \frac{1}{2} x + 1\)[/tex]
3. [tex]\(y = 2x - 2\)[/tex]
4. [tex]\(y = \frac{1}{2} x + 2 \frac{1}{2}\)[/tex]
To find the equation of the line, we will first need to calculate the slope of the line that passes through these two points. The formula for the slope [tex]\(m\)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Plugging in the coordinates of the given points:
[tex]\[ x_1 = 4, y_1 = 3 \][/tex]
[tex]\[ x_2 = 2, y_2 = 2 \][/tex]
We get:
[tex]\[ m = \frac{2 - 3}{2 - 4} = \frac{-1}{-2} = \frac{1}{2} \][/tex]
So the slope [tex]\(m\)[/tex] of our line is [tex]\(\frac{1}{2}\)[/tex].
Next, we will determine the y-intercept [tex]\(b\)[/tex] of the line. The slope-intercept form of a line is given by the equation:
[tex]\[ y = mx + b \][/tex]
We can use the coordinates of one of the points and the slope to find [tex]\(b\)[/tex]. Using the point [tex]\((4, 3)\)[/tex]:
[tex]\[ 3 = \frac{1}{2}(4) + b \][/tex]
Solving for [tex]\(b\)[/tex]:
[tex]\[ 3 = 2 + b \][/tex]
[tex]\[ b = 3 - 2 \][/tex]
[tex]\[ b = 1 \][/tex]
So, the equation of the line is:
[tex]\[ y = \frac{1}{2}x + 1 \][/tex]
Now, we compare this result with the given options:
- Option 1: [tex]\(y = \frac{1}{2} x + 2\)[/tex] => Slope = [tex]\(\frac{1}{2}\)[/tex], y-intercept = 2 (does not match)
- Option 2: [tex]\(y = \frac{1}{2} x + 1\)[/tex] => Slope = [tex]\(\frac{1}{2}\)[/tex], y-intercept = 1 (matches)
- Option 3: [tex]\(y = 2x - 2\)[/tex] => Slope = 2, y-intercept = -2 (does not match)
- Option 4: [tex]\(y = \frac{1}{2} x + 2 \frac{1}{2}\)[/tex] => Slope = [tex]\(\frac{1}{2}\)[/tex], y-intercept = 2.5 (does not match)
Therefore, the correct equation of the line passing through [tex]\((4, 3)\)[/tex] and [tex]\((2, 2)\)[/tex] is:
[tex]\[ y = \frac{1}{2} x + 1 \][/tex]
The matching option is:
[tex]\[ y = \frac{1}{2} x + 1 \][/tex]
