Join the IDNLearn.com community and start exploring a world of knowledge today. Get the information you need from our community of experts, who provide detailed and trustworthy answers.
Sagot :
To determine the equation of a line perpendicular to [tex]\( y + 5 = \frac{2}{3}(x - 11) \)[/tex] that passes through the point [tex]\( (5, -6) \)[/tex], we can follow a detailed, step-by-step process:
1. Rewrite the Given Line in Slope-Intercept Form:
- We need to express the given line in the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- Start with the given equation: [tex]\( y + 5 = \frac{2}{3}(x - 11) \)[/tex].
- Distribute [tex]\( \frac{2}{3} \)[/tex]: [tex]\( y + 5 = \frac{2}{3}x - \frac{2}{3} \cdot 11 \)[/tex].
- Simplify: [tex]\( y + 5 = \frac{2}{3}x - \frac{22}{3} \)[/tex].
- Isolate [tex]\( y \)[/tex]: [tex]\( y = \frac{2}{3}x - \frac{22}{3} - 5 \)[/tex].
- Convert 5 to a fraction: [tex]\( y = \frac{2}{3}x - \frac{22}{3} - \frac{15}{3} \)[/tex].
- Combine like terms: [tex]\( y = \frac{2}{3}x - \frac{37}{3} \)[/tex].
Thus, the slope [tex]\( m \)[/tex] of the given line is [tex]\( \frac{2}{3} \)[/tex].
2. Determine the Perpendicular Slope:
- The slope of a line perpendicular to another is the negative reciprocal of the original slope.
- Original slope [tex]\( m = \frac{2}{3} \)[/tex].
- Perpendicular slope [tex]\( m_{\perp} = -\frac{1}{m} = -\frac{1}{\frac{2}{3}} = -\frac{3}{2} \)[/tex].
3. Using Point-Slope Form to Find the Equation:
- The point-slope form of a line is [tex]\( y - y_1 = m(x - x_1) \)[/tex], where [tex]\( (x_1, y_1) \)[/tex] is a given point on the line and [tex]\( m \)[/tex] is the slope.
- Given point [tex]\( (5, -6) \)[/tex], and perpendicular slope [tex]\( -\frac{3}{2} \)[/tex].
- Substitute into the point-slope form: [tex]\( y - (-6) = -\frac{3}{2}(x - 5) \)[/tex].
- Simplify: [tex]\( y + 6 = -\frac{3}{2}(x - 5) \)[/tex].
- Distribute the slope: [tex]\( y + 6 = -\frac{3}{2}x + \frac{15}{2} \)[/tex].
- Isolate [tex]\( y \)[/tex]: [tex]\( y = -\frac{3}{2}x + \frac{15}{2} - 6 \)[/tex].
- Convert 6 to a fraction: [tex]\( y = -\frac{3}{2}x + \frac{15}{2} - \frac{12}{2} \)[/tex].
- Combine like terms: [tex]\( y = -\frac{3}{2}x + \frac{3}{2} \)[/tex].
Thus, the equation of the line in slope-intercept form ([tex]\( y = mx + b \)[/tex]) that is perpendicular to [tex]\( y + 5 = \frac{2}{3}(x - 11) \)[/tex] and passes through the point [tex]\( (5, -6) \)[/tex] is:
[tex]\[ y = -\frac{3}{2}x + \frac{3}{2} \][/tex]
This matches the option [tex]\( y = -\frac{3}{2}x + \frac{3}{2} \)[/tex].
1. Rewrite the Given Line in Slope-Intercept Form:
- We need to express the given line in the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- Start with the given equation: [tex]\( y + 5 = \frac{2}{3}(x - 11) \)[/tex].
- Distribute [tex]\( \frac{2}{3} \)[/tex]: [tex]\( y + 5 = \frac{2}{3}x - \frac{2}{3} \cdot 11 \)[/tex].
- Simplify: [tex]\( y + 5 = \frac{2}{3}x - \frac{22}{3} \)[/tex].
- Isolate [tex]\( y \)[/tex]: [tex]\( y = \frac{2}{3}x - \frac{22}{3} - 5 \)[/tex].
- Convert 5 to a fraction: [tex]\( y = \frac{2}{3}x - \frac{22}{3} - \frac{15}{3} \)[/tex].
- Combine like terms: [tex]\( y = \frac{2}{3}x - \frac{37}{3} \)[/tex].
Thus, the slope [tex]\( m \)[/tex] of the given line is [tex]\( \frac{2}{3} \)[/tex].
2. Determine the Perpendicular Slope:
- The slope of a line perpendicular to another is the negative reciprocal of the original slope.
- Original slope [tex]\( m = \frac{2}{3} \)[/tex].
- Perpendicular slope [tex]\( m_{\perp} = -\frac{1}{m} = -\frac{1}{\frac{2}{3}} = -\frac{3}{2} \)[/tex].
3. Using Point-Slope Form to Find the Equation:
- The point-slope form of a line is [tex]\( y - y_1 = m(x - x_1) \)[/tex], where [tex]\( (x_1, y_1) \)[/tex] is a given point on the line and [tex]\( m \)[/tex] is the slope.
- Given point [tex]\( (5, -6) \)[/tex], and perpendicular slope [tex]\( -\frac{3}{2} \)[/tex].
- Substitute into the point-slope form: [tex]\( y - (-6) = -\frac{3}{2}(x - 5) \)[/tex].
- Simplify: [tex]\( y + 6 = -\frac{3}{2}(x - 5) \)[/tex].
- Distribute the slope: [tex]\( y + 6 = -\frac{3}{2}x + \frac{15}{2} \)[/tex].
- Isolate [tex]\( y \)[/tex]: [tex]\( y = -\frac{3}{2}x + \frac{15}{2} - 6 \)[/tex].
- Convert 6 to a fraction: [tex]\( y = -\frac{3}{2}x + \frac{15}{2} - \frac{12}{2} \)[/tex].
- Combine like terms: [tex]\( y = -\frac{3}{2}x + \frac{3}{2} \)[/tex].
Thus, the equation of the line in slope-intercept form ([tex]\( y = mx + b \)[/tex]) that is perpendicular to [tex]\( y + 5 = \frac{2}{3}(x - 11) \)[/tex] and passes through the point [tex]\( (5, -6) \)[/tex] is:
[tex]\[ y = -\frac{3}{2}x + \frac{3}{2} \][/tex]
This matches the option [tex]\( y = -\frac{3}{2}x + \frac{3}{2} \)[/tex].
We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. For dependable answers, trust IDNLearn.com. Thank you for visiting, and we look forward to assisting you again.
