Answer :
To determine which equations represent a line that is perpendicular to the line given by [tex]\( 5x - 2y = -6 \)[/tex] and passes through the point [tex]\( (5, -4) \)[/tex], we need to follow several steps:
1. Find the slope of the given line.
The equation of the line can be written in the form [tex]\( Ax + By = C \)[/tex]. For the given line [tex]\( 5x - 2y = -6 \)[/tex], we have:
- [tex]\( A = 5 \)[/tex]
- [tex]\( B = -2 \)[/tex]
The slope [tex]\( m \)[/tex] of the line is given by:
[tex]\[ m = -\frac{A}{B} = -\frac{5}{-2} = \frac{5}{2} \][/tex]
2. Determine the slope of the perpendicular line.
The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope. Therefore, the slope of the perpendicular line is:
[tex]\[ m_{\text{perpendicular}} = -\frac{1}{\frac{5}{2}} = -\frac{2}{5} \][/tex]
3. Check which of the given equations represent lines with this slope and pass through the point [tex]\( (5, -4) \)[/tex].
- Option 1: [tex]\( y = -\frac{2}{5}x - 2 \)[/tex]
- The slope of this line is [tex]\( -\frac{2}{5} \)[/tex].
- Substitute [tex]\( x = 5 \)[/tex] and [tex]\( y = -4 \)[/tex] to verify if the point lies on the line:
[tex]\[ -4 = -\frac{2}{5}(5) - 2 \quad \Rightarrow \quad -4 = -2 - 2 \quad \Rightarrow \quad -4 = -4 \quad (\text{True}) \][/tex]
- This equation is correct.
- Option 2: [tex]\( 2x + 5y = -10 \)[/tex]
- Convert to slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[ 5y = -2x - 10 \quad \Rightarrow \quad y = -\frac{2}{5}x - 2 \][/tex]
- The slope of this line is [tex]\( -\frac{2}{5} \)[/tex].
- Substitute [tex]\( x = 5 \)[/tex] and [tex]\( y = -4 \)[/tex] to verify:
[tex]\[ -4 = -\frac{2}{5}(5) - 2 \quad \Rightarrow \quad -4 = -2 - 2 \quad \Rightarrow \quad -4 = -4 \quad (\text{True}) \][/tex]
- This equation is correct.
- Option 3: [tex]\( 2x - 5y = -10 \)[/tex]
- Convert to slope-intercept form:
[tex]\[ -5y = -2x - 10 \quad \Rightarrow \quad y = \frac{2}{5}x + 2 \][/tex]
- The slope of this line is [tex]\( \frac{2}{5} \)[/tex], not [tex]\( -\frac{2}{5} \)[/tex].
- This equation is not correct.
- Option 4: [tex]\( y + 4 = \frac{2}{5}(x - 5) \)[/tex]
- Convert to slope-intercept form:
[tex]\[ y + 4 = \frac{2}{5}x - \frac{2}{5}(5) \quad \Rightarrow \quad y = \frac{2}{5}x - 2 - 4 \quad \Rightarrow \quad y = \frac{2}{5}x - 6 \][/tex]
- The slope of this line is [tex]\( \frac{2}{5} \)[/tex], not [tex]\( -\frac{2}{5} \)[/tex].
- This equation is not correct.
- Option 5: [tex]\( y - 4 = \frac{5}{2}(x + 5) \)[/tex]
- Convert to slope-intercept form:
[tex]\[ y - 4 = \frac{5}{2}x + \frac{5}{2}(5) \quad \Rightarrow \quad y = \frac{5}{2}x + \frac{25}{2} + 4 \quad \Rightarrow \quad y = \frac{5}{2}x + \frac{33}{2} \][/tex]
- The slope of this line is [tex]\( \frac{5}{2} \)[/tex], not [tex]\( -\frac{2}{5} \)[/tex].
- This equation is not correct.
From the calculations, the correct equations representing the line that is perpendicular to [tex]\( 5x - 2y = -6 \)[/tex] and passing through [tex]\( (5, -4) \)[/tex] are:
- [tex]\( y = -\frac{2}{5}x - 2 \)[/tex]
- [tex]\( 2x + 5y = -10 \)[/tex]
1. Find the slope of the given line.
The equation of the line can be written in the form [tex]\( Ax + By = C \)[/tex]. For the given line [tex]\( 5x - 2y = -6 \)[/tex], we have:
- [tex]\( A = 5 \)[/tex]
- [tex]\( B = -2 \)[/tex]
The slope [tex]\( m \)[/tex] of the line is given by:
[tex]\[ m = -\frac{A}{B} = -\frac{5}{-2} = \frac{5}{2} \][/tex]
2. Determine the slope of the perpendicular line.
The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope. Therefore, the slope of the perpendicular line is:
[tex]\[ m_{\text{perpendicular}} = -\frac{1}{\frac{5}{2}} = -\frac{2}{5} \][/tex]
3. Check which of the given equations represent lines with this slope and pass through the point [tex]\( (5, -4) \)[/tex].
- Option 1: [tex]\( y = -\frac{2}{5}x - 2 \)[/tex]
- The slope of this line is [tex]\( -\frac{2}{5} \)[/tex].
- Substitute [tex]\( x = 5 \)[/tex] and [tex]\( y = -4 \)[/tex] to verify if the point lies on the line:
[tex]\[ -4 = -\frac{2}{5}(5) - 2 \quad \Rightarrow \quad -4 = -2 - 2 \quad \Rightarrow \quad -4 = -4 \quad (\text{True}) \][/tex]
- This equation is correct.
- Option 2: [tex]\( 2x + 5y = -10 \)[/tex]
- Convert to slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[ 5y = -2x - 10 \quad \Rightarrow \quad y = -\frac{2}{5}x - 2 \][/tex]
- The slope of this line is [tex]\( -\frac{2}{5} \)[/tex].
- Substitute [tex]\( x = 5 \)[/tex] and [tex]\( y = -4 \)[/tex] to verify:
[tex]\[ -4 = -\frac{2}{5}(5) - 2 \quad \Rightarrow \quad -4 = -2 - 2 \quad \Rightarrow \quad -4 = -4 \quad (\text{True}) \][/tex]
- This equation is correct.
- Option 3: [tex]\( 2x - 5y = -10 \)[/tex]
- Convert to slope-intercept form:
[tex]\[ -5y = -2x - 10 \quad \Rightarrow \quad y = \frac{2}{5}x + 2 \][/tex]
- The slope of this line is [tex]\( \frac{2}{5} \)[/tex], not [tex]\( -\frac{2}{5} \)[/tex].
- This equation is not correct.
- Option 4: [tex]\( y + 4 = \frac{2}{5}(x - 5) \)[/tex]
- Convert to slope-intercept form:
[tex]\[ y + 4 = \frac{2}{5}x - \frac{2}{5}(5) \quad \Rightarrow \quad y = \frac{2}{5}x - 2 - 4 \quad \Rightarrow \quad y = \frac{2}{5}x - 6 \][/tex]
- The slope of this line is [tex]\( \frac{2}{5} \)[/tex], not [tex]\( -\frac{2}{5} \)[/tex].
- This equation is not correct.
- Option 5: [tex]\( y - 4 = \frac{5}{2}(x + 5) \)[/tex]
- Convert to slope-intercept form:
[tex]\[ y - 4 = \frac{5}{2}x + \frac{5}{2}(5) \quad \Rightarrow \quad y = \frac{5}{2}x + \frac{25}{2} + 4 \quad \Rightarrow \quad y = \frac{5}{2}x + \frac{33}{2} \][/tex]
- The slope of this line is [tex]\( \frac{5}{2} \)[/tex], not [tex]\( -\frac{2}{5} \)[/tex].
- This equation is not correct.
From the calculations, the correct equations representing the line that is perpendicular to [tex]\( 5x - 2y = -6 \)[/tex] and passing through [tex]\( (5, -4) \)[/tex] are:
- [tex]\( y = -\frac{2}{5}x - 2 \)[/tex]
- [tex]\( 2x + 5y = -10 \)[/tex]
