To solve this problem, we need to find the equation of the second line which is parallel to the first line and then find its x-intercept. Here is the detailed, step-by-step solution:
1. Determine the slope of the first line:
The first line passes through the points [tex]\((-1, -3)\)[/tex] and [tex]\((1, 5)\)[/tex]. The formula for the slope ([tex]\(m\)[/tex]) of a line passing through 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]
Substituting the given points:
[tex]\[
m = \frac{5 - (-3)}{1 - (-1)} = \frac{5 + 3}{1 + 1} = \frac{8}{2} = 4
\][/tex]
Therefore, the slope of the first line is [tex]\(4\)[/tex].
2. Form the equation of the second line:
Since the second line is parallel to the first, it has the same slope. Hence, the slope of the second line is also [tex]\(4\)[/tex].
The second line passes through the point [tex]\((2, 3)\)[/tex]. Using the slope-intercept form of the equation of a line [tex]\(y = mx + b\)[/tex], we can find the y-intercept [tex]\(b\)[/tex]:
[tex]\[
3 = 4 \cdot 2 + b
\][/tex]
Solving for [tex]\(b\)[/tex]:
[tex]\[
3 = 8 + b \implies b = 3 - 8 = -5
\][/tex]
Thus, the equation of the second line is:
[tex]\[
y = 4x - 5
\][/tex]
3. Find the x-intercept of the second line:
The x-intercept is the point where the line crosses the x-axis (where [tex]\(y = 0\)[/tex]). Substituting [tex]\(y = 0\)[/tex] in the equation [tex]\(y = 4x - 5\)[/tex]:
[tex]\[
0 = 4x - 5
\][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[
4x = 5 \implies x = \frac{5}{4} = 1.25
\][/tex]
Therefore, the x-intercept of the second line is:
[tex]\[
(1.25, 0)
\][/tex]
So, the correct answer is b. [tex]$(1.25; 0)$[/tex].
