Sure! Let's derive the equation of the line step-by-step.
1. Identify the given information:
- A point on the line: [tex]\((2, 9)\)[/tex]
- The slope (m) of the line: [tex]\(4\)[/tex]
2. Write the equation of the line in slope-intercept form:
The slope-intercept form of a line’s equation is given by:
[tex]\[
y = mx + c
\][/tex]
Where:
- [tex]\(y\)[/tex] is the dependent variable,
- [tex]\(x\)[/tex] is the independent variable,
- [tex]\(m\)[/tex] is the slope of the line,
- [tex]\(c\)[/tex] is the y-intercept of the line.
3. Substitute the known slope and point into the equation to solve for [tex]\(c\)[/tex]:
Given the point [tex]\((2, 9)\)[/tex] and slope [tex]\(m = 4\)[/tex], substituting [tex]\(x = 2\)[/tex], [tex]\(y = 9\)[/tex], and [tex]\(m = 4\)[/tex] into the slope-intercept form:
[tex]\[
9 = 4(2) + c
\][/tex]
4. Solve for [tex]\(c\)[/tex]:
[tex]\[
9 = 8 + c
\][/tex]
[tex]\[
c = 9 - 8
\][/tex]
[tex]\[
c = 1
\][/tex]
5. Write the final equation for the line:
Now that we have the slope [tex]\(m = 4\)[/tex] and the y-intercept [tex]\(c = 1\)[/tex], we can write the equation of the line:
[tex]\[
y = 4x + 1
\][/tex]
So, the equation of the line that passes through the point [tex]\((2, 9)\)[/tex] with a slope of [tex]\(4\)[/tex] is:
[tex]\[
y = 4x + 1
\][/tex]
