IDNLearn.com offers a unique blend of expert answers and community-driven insights. Get the information you need from our community of experts, who provide detailed and trustworthy answers.
Sagot :
To find the equation of a line that is parallel to the given line [tex]\( y = \frac{6}{5}x + 10 \)[/tex] and passes through the point [tex]\( (12, -2) \)[/tex], we need to follow these steps:
1. Determine the slope of the given line:
The slope-intercept form of a line is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] represents the slope. For the given line [tex]\( y = \frac{6}{5}x + 10 \)[/tex], the slope [tex]\( m \)[/tex] is [tex]\( \frac{6}{5} \)[/tex].
2. Understand parallel lines:
Lines that are parallel have the same slope. Therefore, the slope of the line that we need to find is also [tex]\( \frac{6}{5} \)[/tex].
3. Use the point-slope form of the line:
The point-slope form of a line passing through a point [tex]\( (x_1, y_1) \)[/tex] with slope [tex]\( m \)[/tex] is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Plugging in the coordinates [tex]\( (12, -2) \)[/tex] and the slope [tex]\( \frac{6}{5} \)[/tex]:
[tex]\[ y - (-2) = \frac{6}{5}(x - 12) \][/tex]
Simplifying:
[tex]\[ y + 2 = \frac{6}{5}x - \frac{6}{5} \cdot 12 \][/tex]
[tex]\[ y + 2 = \frac{6}{5}x - \frac{72}{5} \][/tex]
4. Isolate [tex]\( y \)[/tex]:
Subtract 2 from both sides to solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{6}{5}x - \frac{72}{5} - 2 \][/tex]
To combine the constants, express 2 as a fraction with the same denominator:
[tex]\[ 2 = \frac{10}{5} \][/tex]
Therefore,
[tex]\[ y = \frac{6}{5}x - \frac{72}{5} - \frac{10}{5} \][/tex]
[tex]\[ y = \frac{6}{5}x - \frac{82}{5} \][/tex]
So, the equation of the line that is parallel to the given line [tex]\( y = \frac{6}{5}x + 10 \)[/tex] and passes through the point [tex]\( (12, -2) \)[/tex] is:
[tex]\[ y = \frac{6}{5}x - \frac{82}{5} \][/tex]
To convert [tex]\( \frac{82}{5} \)[/tex] to a decimal:
[tex]\[ \frac{82}{5} = 16.4 \][/tex]
Thus, the final equation is:
[tex]\[ y = \frac{6}{5}x - 16.4 \][/tex]
This equation matches none of the provided options exactly. The closest matching option provided is [tex]\( y = \frac{6}{5}x - 10 \)[/tex], but that's incorrect based on our step-by-step derivation. The correct option is not provided explicitly in the options list. The result from the process confirms our calculated y-intercept and slope.
Nonetheless, based on the derived correct result, the accurate equation is:
[tex]\[ y = \frac{6}{5}x - 16.4 \][/tex]
This should be the final correct equation for the question.
1. Determine the slope of the given line:
The slope-intercept form of a line is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] represents the slope. For the given line [tex]\( y = \frac{6}{5}x + 10 \)[/tex], the slope [tex]\( m \)[/tex] is [tex]\( \frac{6}{5} \)[/tex].
2. Understand parallel lines:
Lines that are parallel have the same slope. Therefore, the slope of the line that we need to find is also [tex]\( \frac{6}{5} \)[/tex].
3. Use the point-slope form of the line:
The point-slope form of a line passing through a point [tex]\( (x_1, y_1) \)[/tex] with slope [tex]\( m \)[/tex] is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Plugging in the coordinates [tex]\( (12, -2) \)[/tex] and the slope [tex]\( \frac{6}{5} \)[/tex]:
[tex]\[ y - (-2) = \frac{6}{5}(x - 12) \][/tex]
Simplifying:
[tex]\[ y + 2 = \frac{6}{5}x - \frac{6}{5} \cdot 12 \][/tex]
[tex]\[ y + 2 = \frac{6}{5}x - \frac{72}{5} \][/tex]
4. Isolate [tex]\( y \)[/tex]:
Subtract 2 from both sides to solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{6}{5}x - \frac{72}{5} - 2 \][/tex]
To combine the constants, express 2 as a fraction with the same denominator:
[tex]\[ 2 = \frac{10}{5} \][/tex]
Therefore,
[tex]\[ y = \frac{6}{5}x - \frac{72}{5} - \frac{10}{5} \][/tex]
[tex]\[ y = \frac{6}{5}x - \frac{82}{5} \][/tex]
So, the equation of the line that is parallel to the given line [tex]\( y = \frac{6}{5}x + 10 \)[/tex] and passes through the point [tex]\( (12, -2) \)[/tex] is:
[tex]\[ y = \frac{6}{5}x - \frac{82}{5} \][/tex]
To convert [tex]\( \frac{82}{5} \)[/tex] to a decimal:
[tex]\[ \frac{82}{5} = 16.4 \][/tex]
Thus, the final equation is:
[tex]\[ y = \frac{6}{5}x - 16.4 \][/tex]
This equation matches none of the provided options exactly. The closest matching option provided is [tex]\( y = \frac{6}{5}x - 10 \)[/tex], but that's incorrect based on our step-by-step derivation. The correct option is not provided explicitly in the options list. The result from the process confirms our calculated y-intercept and slope.
Nonetheless, based on the derived correct result, the accurate equation is:
[tex]\[ y = \frac{6}{5}x - 16.4 \][/tex]
This should be the final correct equation for the question.
Thank you for using this platform to share and learn. Don't hesitate to keep asking and answering. We value every contribution you make. IDNLearn.com provides the best answers to your questions. Thank you for visiting, and come back soon for more helpful information.
