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Which equations represent the line that is perpendicular to the line [tex]5x - 2y = -6[/tex] and passes through the point [tex]\((5, -4)\)[/tex]? Select three options.

A. [tex]y = -\frac{2}{5}x - 2[/tex]
B. [tex]2x + 5y = -10[/tex]
C. [tex]2x - 5y = -10[/tex]
D. [tex]y + 4 = -\frac{2}{5}(x - 5)[/tex]
E. [tex]y - 4 = \frac{5}{2}(x + 5)[/tex]

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.
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