To write the equation of a line passing through the point [tex]\((1, 4)\)[/tex] with a slope of [tex]\(-7\)[/tex] in the [tex]\(Ax + By = C\)[/tex] form, follow these steps:
1. Identify the given information:
- The point the line passes through: [tex]\((1, 4)\)[/tex]
- The slope of the line: [tex]\(m = -7\)[/tex]
2. Write the equation in slope-intercept form [tex]\(y = mx + b\)[/tex]:
- Using the slope [tex]\(m\)[/tex] and the point [tex]\((x_1, y_1)\)[/tex], start with the slope-intercept form of a line:
[tex]\[
y = mx + b
\][/tex]
- Substitute the slope [tex]\(m = -7\)[/tex]:
[tex]\[
y = -7x + b
\][/tex]
3. Find the y-intercept [tex]\(b\)[/tex]:
- Substitute the given point [tex]\((1, 4)\)[/tex] into the equation [tex]\(y = -7x + b\)[/tex] to solve for [tex]\(b\)[/tex]:
[tex]\[
4 = -7(1) + b
\][/tex]
[tex]\[
4 = -7 + b
\][/tex]
[tex]\[
b = 11
\][/tex]
4. Write the full slope-intercept form of the line:
- Now that we have the slope and the y-intercept, the equation is:
[tex]\[
y = -7x + 11
\][/tex]
5. Convert the slope-intercept form to the standard form [tex]\(Ax + By = C\)[/tex]:
- Start with the equation [tex]\(y = -7x + 11\)[/tex].
- To convert this to [tex]\(Ax + By = C\)[/tex], rearrange the terms to get all terms involving variables on one side of the equation:
[tex]\[
y + 7x = 11
\][/tex]
- Since the standard form typically has the [tex]\(x\)[/tex]-term first, write:
[tex]\[
7x + y = 11
\][/tex]
6. Identify the coefficients [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex]:
- Comparing [tex]\(7x + y = 11\)[/tex] with [tex]\(Ax + By = C\)[/tex], we get:
[tex]\[
A = 7, \quad B = 1, \quad C = 11
\][/tex]
So, the equation of the line in [tex]\(Ax + By = C\)[/tex] form with integer coefficients is:
[tex]\[
7x + y = 11
\][/tex]
