Answer :
To determine the equation of a line parallel to the line given by [tex]\( y + 5 = \frac{7}{5}(x - 16) \)[/tex] that passes through the point [tex]\((14, 0)\)[/tex], follow these steps:
### Step 1: Convert the Given Line into Slope-Intercept Form
First, we need to convert the equation [tex]\( y + 5 = \frac{7}{5}(x - 16) \)[/tex] into the slope-intercept form, [tex]\( y = mx + b \)[/tex].
1. Distribute the [tex]\(\frac{7}{5}\)[/tex] on the right-hand side:
[tex]\[ y + 5 = \frac{7}{5}x - \frac{7}{5} \cdot 16 \][/tex]
2. Simplify the constants:
[tex]\[ y + 5 = \frac{7}{5}x - \frac{112}{5} \][/tex]
3. Isolate [tex]\( y \)[/tex]:
[tex]\[ y = \frac{7}{5}x - \frac{112}{5} - 5 \][/tex]
4. Combine like terms:
[tex]\[ y = \frac{7}{5}x - \left(\frac{112}{5} + \frac{25}{5}\right) \][/tex]
[tex]\[ y = \frac{7}{5}x - \frac{137}{5} \][/tex]
Therefore, the slope [tex]\( m \)[/tex] of the line is [tex]\(\frac{7}{5} \)[/tex].
### Step 2: Use the Point-Slope Form to Find the New Line
Since parallel lines have the same slope, the slope of the new line is also [tex]\(\frac{7}{5}\)[/tex]. We use the point [tex]\((14, 0)\)[/tex] and the point-slope form of a line [tex]\( y - y_1 = m(x - x_1) \)[/tex]:
1. Plugging in the slope [tex]\( m = \frac{7}{5} \)[/tex] and the point [tex]\((14, 0)\)[/tex]:
[tex]\[ y - 0 = \frac{7}{5}(x - 14) \][/tex]
2. Simplifying:
[tex]\[ y = \frac{7}{5}x - \frac{7}{5} \cdot 14 \][/tex]
### Step 3: Calculate the Intercept
1. Simplify the y-intercept calculation:
[tex]\[ y = \frac{7}{5}x - \frac{98}{5} \][/tex]
### Answer
The equation of the new line, parallel to the given line and passing through the point [tex]\((14, 0)\)[/tex], is:
[tex]\[ \boxed{y = \frac{7}{5}x - \frac{98}{5}} \][/tex]
So the correct answer is:
[tex]\[ y = \frac{7}{5}x - \frac{98}{5} \][/tex]
### Step 1: Convert the Given Line into Slope-Intercept Form
First, we need to convert the equation [tex]\( y + 5 = \frac{7}{5}(x - 16) \)[/tex] into the slope-intercept form, [tex]\( y = mx + b \)[/tex].
1. Distribute the [tex]\(\frac{7}{5}\)[/tex] on the right-hand side:
[tex]\[ y + 5 = \frac{7}{5}x - \frac{7}{5} \cdot 16 \][/tex]
2. Simplify the constants:
[tex]\[ y + 5 = \frac{7}{5}x - \frac{112}{5} \][/tex]
3. Isolate [tex]\( y \)[/tex]:
[tex]\[ y = \frac{7}{5}x - \frac{112}{5} - 5 \][/tex]
4. Combine like terms:
[tex]\[ y = \frac{7}{5}x - \left(\frac{112}{5} + \frac{25}{5}\right) \][/tex]
[tex]\[ y = \frac{7}{5}x - \frac{137}{5} \][/tex]
Therefore, the slope [tex]\( m \)[/tex] of the line is [tex]\(\frac{7}{5} \)[/tex].
### Step 2: Use the Point-Slope Form to Find the New Line
Since parallel lines have the same slope, the slope of the new line is also [tex]\(\frac{7}{5}\)[/tex]. We use the point [tex]\((14, 0)\)[/tex] and the point-slope form of a line [tex]\( y - y_1 = m(x - x_1) \)[/tex]:
1. Plugging in the slope [tex]\( m = \frac{7}{5} \)[/tex] and the point [tex]\((14, 0)\)[/tex]:
[tex]\[ y - 0 = \frac{7}{5}(x - 14) \][/tex]
2. Simplifying:
[tex]\[ y = \frac{7}{5}x - \frac{7}{5} \cdot 14 \][/tex]
### Step 3: Calculate the Intercept
1. Simplify the y-intercept calculation:
[tex]\[ y = \frac{7}{5}x - \frac{98}{5} \][/tex]
### Answer
The equation of the new line, parallel to the given line and passing through the point [tex]\((14, 0)\)[/tex], is:
[tex]\[ \boxed{y = \frac{7}{5}x - \frac{98}{5}} \][/tex]
So the correct answer is:
[tex]\[ y = \frac{7}{5}x - \frac{98}{5} \][/tex]
