To determine the equation of a line in standard form that goes through the point [tex]\((2, -1)\)[/tex] and is parallel to another line with a slope of [tex]\(\frac{3}{4}\)[/tex], follow these steps:
1. Identify the given information:
- Point: [tex]\((2, -1)\)[/tex]
- Slope: [tex]\(\frac{3}{4}\)[/tex]
2. Use the point-slope form of the line equation:
The point-slope form of a line is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Substituting the given point [tex]\((2, -1)\)[/tex] and the slope [tex]\(m = \frac{3}{4}\)[/tex]:
[tex]\[
y - (-1) = \frac{3}{4}(x - 2)
\][/tex]
Simplifying this:
[tex]\[
y + 1 = \frac{3}{4}(x - 2)
\][/tex]
3. Distribute the slope on the right-hand side:
[tex]\[
y + 1 = \frac{3}{4}x - \frac{3}{2}
\][/tex]
4. Clear the fraction by multiplying every term by 4:
[tex]\[
4(y + 1) = 3x - 6
\][/tex]
Simplifying this:
[tex]\[
4y + 4 = 3x - 6
\][/tex]
5. Rearrange the equation to the standard form [tex]\(Ax + By = C\)[/tex]:
Move all terms to one side of the equation:
[tex]\[
-3x + 4y = -10
\][/tex]
6. Standard form conventionally requires the coefficient [tex]\(A\)[/tex] to be positive:
Multiply the entire equation by -1 to make [tex]\(A\)[/tex] positive:
[tex]\[
3x - 4y = 10
\][/tex]
Therefore, the equation of the line in standard form is:
[tex]\[
\boxed{3x - 4y = 10}
\][/tex]
