Question

A line passes through the point (4,
–
2) and has a slope of
1
2
. Which shows an equation for this line written in standard form?

Answer 1

To find the equation of a line in standard form, we can start with the point-slope form of a line equation:

\[
y - y_1 = m(x - x_1)
\]

where \((x_1, y_1)\) is a point on the line, and \(m\) is the slope. In this case, \((x_1, y_1) = (4, -2)\) and \(m = \frac{1}{2}\).

Plugging in the values, we have:

\[
y - (-2) = \frac{1}{2}(x - 4)
\]

This simplifies to:

\[
y + 2 = \frac{1}{2}(x - 4)
\]

Next, we can multiply both sides by 2 to eliminate the fraction:

\[
2(y + 2) = x - 4
\]

Which expands to:

\[
2y + 4 = x - 4
\]

Now we rearrange this equation to standard form \(Ax + By = C\). To do this, we move \(x\) and the constants to one side:

\[
- x + 2y = -4 - 4
\]

or

\[
-x + 2y = -8
\]

To write it in standard form with \(A\) being positive, we can multiply through by -1:

\[
x - 2y = 8
\]

Thus, the equation of the line in standard form is:

\[
x - 2y = 8
\]