To find the equation of the line that touches the x-axis at the point (3,0) and passes through the point (1,2), let's work through the calculations step-by-step.
1. Identify the known coordinates:
- Point on the x-axis: [tex]\((3, 0)\)[/tex]
- Another point through which the line passes: [tex]\((1, 2)\)[/tex]
2. Calculate the slope of the line ([tex]\(m\)[/tex]):
The formula for calculating 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]
Substituting the given points [tex]\((x_1, y_1) = (3, 0)\)[/tex] and [tex]\((x_2, y_2) = (1, 2)\)[/tex]:
[tex]\[
m = \frac{2 - 0}{1 - 3} = \frac{2}{-2} = -1
\][/tex]
3. Use the point-slope form to find the equation of the line:
The point-slope form of the line's equation is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Using the point [tex]\((3, 0)\)[/tex] and the slope [tex]\(m = -1\)[/tex]:
[tex]\[
y - 0 = -1(x - 3)
\][/tex]
Simplify this equation to get it into slope-intercept form [tex]\(y = mx + c\)[/tex]:
[tex]\[
y = -1(x - 3)
\][/tex]
[tex]\[
y = -x + 3
\][/tex]
4. Identify the y-intercept (c):
The y-intercept [tex]\(c\)[/tex] is the constant term in the slope-intercept form [tex]\(y = mx + c\)[/tex]. From the equation [tex]\(y = -x + 3\)[/tex], we see that the y-intercept is:
[tex]\[
c = 3
\][/tex]
Therefore, the slope of the line is [tex]\(-1\)[/tex], the y-intercept is [tex]\(3\)[/tex], and the equation of the line is:
[tex]\[
y = -x + 3
\][/tex]
