To convert the given point-slope form equation [tex]\( y - \frac{2}{3} = -2 \left( x - \frac{1}{4} \right) \)[/tex] into slope-intercept form [tex]\( y = mx + b \)[/tex], follow these steps:
1. Distribute the slope on the right side:
[tex]\[
y - \frac{2}{3} = -2 \left( x - \frac{1}{4} \right)
\][/tex]
Expand the right side by distributing [tex]\(-2\)[/tex]:
[tex]\[
y - \frac{2}{3} = -2x + 2 \cdot \frac{1}{4}
\][/tex]
Since [tex]\( 2 \cdot \frac{1}{4} = \frac{2}{4} = \frac{1}{2} \)[/tex], we get:
[tex]\[
y - \frac{2}{3} = -2x + \frac{1}{2}
\][/tex]
2. Isolate [tex]\( y \)[/tex] by adding [tex]\(\frac{2}{3}\)[/tex] to both sides of the equation:
[tex]\[
y = -2x + \frac{1}{2} + \frac{2}{3}
\][/tex]
3. Combine the fractions [tex]\(\frac{1}{2}\)[/tex] and [tex]\(\frac{2}{3}\)[/tex] on the right side:
To combine these fractions, find a common denominator. The common denominator for 2 and 3 is 6. Convert each fraction:
[tex]\[
\frac{1}{2} = \frac{3}{6}, \quad \frac{2}{3} = \frac{4}{6}
\][/tex]
Adding these fractions:
[tex]\[
\frac{1}{2} + \frac{2}{3} = \frac{3}{6} + \frac{4}{6} = \frac{7}{6}
\][/tex]
4. Substitute back into the equation:
[tex]\[
y = -2x + \frac{7}{6}
\][/tex]
Thus, the equation in slope-intercept form is:
[tex]\[
y = -2x + \frac{7}{6}
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{y = -2x + \frac{7}{6}}
\][/tex]
