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To determine which of the given equations represent the line that is perpendicular to [tex]\( 5x - 2y = -6 \)[/tex] and passes through the point [tex]\( (5, -4) \)[/tex], we should follow these steps:
1. Find the slope of the given line:
- The given line is [tex]\( 5x - 2y = -6 \)[/tex].
- To find the slope, we first rewrite this line in slope-intercept form ([tex]\( y = mx + b \)[/tex]) where [tex]\( m \)[/tex] is the slope.
- Solving for [tex]\( y \)[/tex]:
[tex]\[ 5x - 2y = -6 \implies -2y = -5x - 6 \implies y = \frac{5}{2}x + 3 \][/tex]
- So, the slope ([tex]\( m \)[/tex]) of the given line is [tex]\( \frac{5}{2} \)[/tex].
2. Find the slope of the perpendicular line:
- For a line to be perpendicular to another, its slope must be the negative reciprocal of the other line's slope.
- The negative reciprocal of [tex]\( \frac{5}{2} \)[/tex] is [tex]\( -\frac{2}{5} \)[/tex].
- Therefore, the slope of the perpendicular line is [tex]\( -\frac{2}{5} \)[/tex].
3. Use the point-slope form to find the equation of the perpendicular line:
- The perpendicular line passes through the point [tex]\( (5, -4) \)[/tex] and has a slope of [tex]\( -\frac{2}{5} \)[/tex].
- We use the point-slope form [tex]\( y - y_1 = m(x - x_1) \)[/tex]:
[tex]\[ y - (-4) = -\frac{2}{5}(x - 5) \implies y + 4 = -\frac{2}{5}(x - 5) \][/tex]
- This is one of the forms of our answer:
[tex]\[ y + 4 = -\frac{2}{5}(x - 5) \][/tex]
4. Convert the equation to standard form (Ax + By = C):
- Starting with [tex]\( y + 4 = -\frac{2}{5}(x - 5) \)[/tex]:
[tex]\[ y + 4 = -\frac{2}{5}x + 2 \][/tex]
- To clear the fraction, multiply every term by 5:
[tex]\[ 5y + 20 = -2x + 10 \][/tex]
- Rearranging terms to get it in standard form [tex]\( Ax + By = C \)[/tex]:
[tex]\[ 2x + 5y = -10 \][/tex]
- This is another form of our answer:
[tex]\[ 2x + 5y = -10 \][/tex]
5. Check the provided options to see which equations match our forms:
- [tex]\( y = -\frac{2}{5}x - 2 \)[/tex]: This is not correct because it has a different y-intercept and does not pass through the given point.
- [tex]\( 2x + 5y = -10 \)[/tex]: This is correct.
- [tex]\( 2x - 5y = -10 \)[/tex]: This is not correct because it forms a positive reciprocal slope when rearranged to solve for y.
- [tex]\( y + 4 = -\frac{2}{5}(x - 5) \)[/tex]: This is correct.
- [tex]\( y - 4 = \frac{5}{2}(x + 5) \)[/tex]: This is not correct because it forms a positive reciprocal slope which is not perpendicular.
So, the correct equations are:
- [tex]\( 2x + 5y = -10 \)[/tex]
- [tex]\( y + 4 = -\frac{2}{5}(x - 5) \)[/tex]
The options given do not have a third correct equation. The question may be erroneous in asking for three options, as only two valid equations are provided based on our solution steps.
1. Find the slope of the given line:
- The given line is [tex]\( 5x - 2y = -6 \)[/tex].
- To find the slope, we first rewrite this line in slope-intercept form ([tex]\( y = mx + b \)[/tex]) where [tex]\( m \)[/tex] is the slope.
- Solving for [tex]\( y \)[/tex]:
[tex]\[ 5x - 2y = -6 \implies -2y = -5x - 6 \implies y = \frac{5}{2}x + 3 \][/tex]
- So, the slope ([tex]\( m \)[/tex]) of the given line is [tex]\( \frac{5}{2} \)[/tex].
2. Find the slope of the perpendicular line:
- For a line to be perpendicular to another, its slope must be the negative reciprocal of the other line's slope.
- The negative reciprocal of [tex]\( \frac{5}{2} \)[/tex] is [tex]\( -\frac{2}{5} \)[/tex].
- Therefore, the slope of the perpendicular line is [tex]\( -\frac{2}{5} \)[/tex].
3. Use the point-slope form to find the equation of the perpendicular line:
- The perpendicular line passes through the point [tex]\( (5, -4) \)[/tex] and has a slope of [tex]\( -\frac{2}{5} \)[/tex].
- We use the point-slope form [tex]\( y - y_1 = m(x - x_1) \)[/tex]:
[tex]\[ y - (-4) = -\frac{2}{5}(x - 5) \implies y + 4 = -\frac{2}{5}(x - 5) \][/tex]
- This is one of the forms of our answer:
[tex]\[ y + 4 = -\frac{2}{5}(x - 5) \][/tex]
4. Convert the equation to standard form (Ax + By = C):
- Starting with [tex]\( y + 4 = -\frac{2}{5}(x - 5) \)[/tex]:
[tex]\[ y + 4 = -\frac{2}{5}x + 2 \][/tex]
- To clear the fraction, multiply every term by 5:
[tex]\[ 5y + 20 = -2x + 10 \][/tex]
- Rearranging terms to get it in standard form [tex]\( Ax + By = C \)[/tex]:
[tex]\[ 2x + 5y = -10 \][/tex]
- This is another form of our answer:
[tex]\[ 2x + 5y = -10 \][/tex]
5. Check the provided options to see which equations match our forms:
- [tex]\( y = -\frac{2}{5}x - 2 \)[/tex]: This is not correct because it has a different y-intercept and does not pass through the given point.
- [tex]\( 2x + 5y = -10 \)[/tex]: This is correct.
- [tex]\( 2x - 5y = -10 \)[/tex]: This is not correct because it forms a positive reciprocal slope when rearranged to solve for y.
- [tex]\( y + 4 = -\frac{2}{5}(x - 5) \)[/tex]: This is correct.
- [tex]\( y - 4 = \frac{5}{2}(x + 5) \)[/tex]: This is not correct because it forms a positive reciprocal slope which is not perpendicular.
So, the correct equations are:
- [tex]\( 2x + 5y = -10 \)[/tex]
- [tex]\( y + 4 = -\frac{2}{5}(x - 5) \)[/tex]
The options given do not have a third correct equation. The question may be erroneous in asking for three options, as only two valid equations are provided based on our solution steps.
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