Answer :
Certainly! Let's go through the steps to find the equation of a line parallel to the line [tex]\( y = \frac{4}{3}x - 16 \)[/tex] that passes through the point [tex]\((-12, 0)\)[/tex].
### Step-by-Step Solution:
1. Identify the Slope:
- The original line is given by [tex]\( y = \frac{4}{3}x - 16 \)[/tex].
- The slope-intercept form of a line is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope.
- From the given line equation, the slope ([tex]\( m \)[/tex]) is [tex]\( \frac{4}{3} \)[/tex].
2. Understand Parallel Lines:
- Lines that are parallel have identical slopes.
- Therefore, the slope of the parallel line is also [tex]\( \frac{4}{3} \)[/tex].
3. Find the Y-Intercept of the New Line:
- The new line must pass through the point [tex]\((-12, 0)\)[/tex].
- Using the point-slope form of the line equation [tex]\( y = mx + b \)[/tex], we will substitute the given point [tex]\((-12, 0)\)[/tex] into the equation to find the y-intercept ([tex]\( b \)[/tex]).
- So, [tex]\( y = \frac{4}{3}x + b \)[/tex].
- Substitute [tex]\( x = -12 \)[/tex] and [tex]\( y = 0 \)[/tex]:
[tex]\[ 0 = \frac{4}{3}(-12) + b \][/tex]
4. Solve for [tex]\( b \)[/tex]:
- Calculate [tex]\( \frac{4}{3} \times (-12) \)[/tex]:
[tex]\[ \frac{4}{3} \times (-12) = -16 \][/tex]
- Now, solve for [tex]\( b \)[/tex]:
[tex]\[ 0 = -16 + b \][/tex]
[tex]\[ b = 16 \][/tex]
5. Write the Equation of the New Line:
- Now we know the slope ([tex]\( \frac{4}{3} \)[/tex]) and the y-intercept ([tex]\( 16 \)[/tex]).
- Therefore, the equation of the new line is:
[tex]\[ y = \frac{4}{3}x + 16 \][/tex]
Among the provided options, none match the correct equation. Thus, the equation of the line parallel to [tex]\( y = \frac{4}{3}x - 16 \)[/tex] and passing through [tex]\((-12, 0)\)[/tex] is [tex]\( y = \frac{4}{3}x + 16 \)[/tex].
### Step-by-Step Solution:
1. Identify the Slope:
- The original line is given by [tex]\( y = \frac{4}{3}x - 16 \)[/tex].
- The slope-intercept form of a line is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope.
- From the given line equation, the slope ([tex]\( m \)[/tex]) is [tex]\( \frac{4}{3} \)[/tex].
2. Understand Parallel Lines:
- Lines that are parallel have identical slopes.
- Therefore, the slope of the parallel line is also [tex]\( \frac{4}{3} \)[/tex].
3. Find the Y-Intercept of the New Line:
- The new line must pass through the point [tex]\((-12, 0)\)[/tex].
- Using the point-slope form of the line equation [tex]\( y = mx + b \)[/tex], we will substitute the given point [tex]\((-12, 0)\)[/tex] into the equation to find the y-intercept ([tex]\( b \)[/tex]).
- So, [tex]\( y = \frac{4}{3}x + b \)[/tex].
- Substitute [tex]\( x = -12 \)[/tex] and [tex]\( y = 0 \)[/tex]:
[tex]\[ 0 = \frac{4}{3}(-12) + b \][/tex]
4. Solve for [tex]\( b \)[/tex]:
- Calculate [tex]\( \frac{4}{3} \times (-12) \)[/tex]:
[tex]\[ \frac{4}{3} \times (-12) = -16 \][/tex]
- Now, solve for [tex]\( b \)[/tex]:
[tex]\[ 0 = -16 + b \][/tex]
[tex]\[ b = 16 \][/tex]
5. Write the Equation of the New Line:
- Now we know the slope ([tex]\( \frac{4}{3} \)[/tex]) and the y-intercept ([tex]\( 16 \)[/tex]).
- Therefore, the equation of the new line is:
[tex]\[ y = \frac{4}{3}x + 16 \][/tex]
Among the provided options, none match the correct equation. Thus, the equation of the line parallel to [tex]\( y = \frac{4}{3}x - 16 \)[/tex] and passing through [tex]\((-12, 0)\)[/tex] is [tex]\( y = \frac{4}{3}x + 16 \)[/tex].
