The equation [tex]$y-\frac{2}{3}=-2\left(x-\frac{1}{4}\right)$[/tex] is written in point-slope form. What is the equation in slope-intercept form?

A. [tex]y=-2x-\frac{1}{6}[/tex]
B. [tex]y=-2x+\frac{7}{6}[/tex]
C. [tex]y=-2x+\frac{5}{12}[/tex]
D. [tex]y=-2x-\frac{11}{12}[/tex]



Answer :

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]

Answer:

B. y=-2x+7/6

Step-by-step explanation:

y - 2/3 = -2 ( x-1/4)

Slope intercept form is y = mx+b  where m is the slope and b is the y intercept.

Distribute the 2.

y -2/3 = -2x + 1/2

Add 2/3 to each side.

y - 2/3 + 2/3 = -2x + 1/2 + 2/3

y = -2x+ 1/2 + 2/3

Get a common denominator of 6.

y = -2x + 3/6 + 4/6

y = -2x+ 7/6