Discover a wealth of knowledge at Westonci.ca, where experts provide answers to your most pressing questions. Join our Q&A platform to connect with experts dedicated to providing precise answers to your questions in different areas. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
Sagot :
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.
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.
