To find an equation of a line parallel to the given line [tex]\(EF\)[/tex] that passes through the point [tex]\((2, 6)\)[/tex], let's follow these steps:
### Step 1: Identify the Slope of the Given Line EF
The line EF has its equation given in slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept. In this case, the slope [tex]\( m \)[/tex] is [tex]\(-\frac{2}{3}\)[/tex].
### Step 2: Understand Parallel Lines
For two lines to be parallel, they must have the same slope. Therefore, the slope of the new line we are finding will also be [tex]\(-\frac{2}{3}\)[/tex].
### Step 3: Use the Point-Slope Form of the Line Equation
The point-slope form of the equation of a line is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\( (x_1, y_1) \)[/tex] is a point on the line and [tex]\( m \)[/tex] is the slope.
Given:
- Slope [tex]\( m = -\frac{2}{3} \)[/tex]
- Point [tex]\( (x_1, y_1) = (2, 6) \)[/tex]
### Step 4: Substitute the Values into the Point-Slope Form
Substitute [tex]\( x_1 = 2 \)[/tex], [tex]\( y_1 = 6 \)[/tex], and [tex]\( m = -\frac{2}{3} \)[/tex] into the point-slope form:
[tex]\[ y - 6 = -\frac{2}{3}(x - 2) \][/tex]
### Step 5: Simplify to Get the Slope-Intercept Form
Simplify the equation to express it in the [tex]\( y = mx + b \)[/tex] form:
[tex]\[ y - 6 = -\frac{2}{3}(x - 2) \][/tex]
[tex]\[ y - 6 = -\frac{2}{3}x + \frac{4}{3} \][/tex]
[tex]\[ y = -\frac{2}{3}x + \frac{4}{3} + 6 \][/tex]
### Step 6: Combine Like Terms
Convert 6 to a fraction with a common denominator of 3:
[tex]\[ 6 = \frac{18}{3} \][/tex]
Add the fractions:
[tex]\[ y = -\frac{2}{3}x + \frac{4}{3} + \frac{18}{3} \][/tex]
[tex]\[ y = -\frac{2}{3}x + \frac{22}{3} \][/tex]
### Step 7: Final Equation
The equation of the line in slope-intercept form that is parallel to the line [tex]\(EF\)[/tex] and passes through the point [tex]\((2, 6)\)[/tex] is:
[tex]\[ y = -\frac{2}{3}x + \frac{22}{3} \][/tex]
Therefore, the correct answer is:
[tex]\[ y = -\frac{2}{3}x + \frac{22}{3} \][/tex]
