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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.
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