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Sagot :
To find the equation of the line that is parallel to a given line and passes through a specific point, we need to follow these steps:
1. Identify the slope of the given line. Lines that are parallel have the same slope.
2. Use the point-slope form of the equation of a line. This form is useful when we know a point through which the line passes and its slope.
First, let’s find the slope of the given line [tex]\(3x - 4y = -17\)[/tex].
1. Rearrange the equation in the slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope:
[tex]\[ 3x - 4y = -17 \][/tex]
[tex]\[ -4y = -3x - 17 \][/tex]
[tex]\[ y = \frac{3}{4}x + \frac{17}{4} \][/tex]
From this, we see that the slope [tex]\(m\)[/tex] is [tex]\(\frac{3}{4}\)[/tex].
Now, we use the point-slope form of the equation of a line [tex]\(y - y_1 = m(x - x_1)\)[/tex], where [tex]\((x_1, y_1)\)[/tex] is a point on the line. In this case, the point is [tex]\((-3, 2)\)[/tex].
[tex]\[ y - 2 = \frac{3}{4}(x + 3) \][/tex]
We can simplify this equation:
[tex]\[ y - 2 = \frac{3}{4}x + \frac{3}{4}(3) \][/tex]
[tex]\[ y - 2 = \frac{3}{4}x + \frac{9}{4} \][/tex]
[tex]\[ y = \frac{3}{4}x + \frac{9}{4} + 2 \][/tex]
[tex]\[ y = \frac{3}{4}x + \frac{9}{4} + \frac{8}{4} \][/tex]
[tex]\[ y = \frac{3}{4}x + \frac{17}{4} \][/tex]
Now we convert this back to the standard form [tex]\(Ax + By = C\)[/tex]:
1. Multiply every term by 4 to clear the fractions:
[tex]\[ 4y = 3x + 17 \][/tex]
[tex]\[ 3x - 4y = -17 \][/tex]
So, the simplified equation of the line parallel to [tex]\(3x - 4y = -17\)[/tex] that passes through the point [tex]\((-3, 2)\)[/tex] turns out to be simpler than expected.
Thus, the required line equation is [tex]\(3x - 4y = -17\)[/tex].
However exact result may take is self-reflection based on presumed passed points and following accordance of option may infer refined calculations.
1. Identify the slope of the given line. Lines that are parallel have the same slope.
2. Use the point-slope form of the equation of a line. This form is useful when we know a point through which the line passes and its slope.
First, let’s find the slope of the given line [tex]\(3x - 4y = -17\)[/tex].
1. Rearrange the equation in the slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope:
[tex]\[ 3x - 4y = -17 \][/tex]
[tex]\[ -4y = -3x - 17 \][/tex]
[tex]\[ y = \frac{3}{4}x + \frac{17}{4} \][/tex]
From this, we see that the slope [tex]\(m\)[/tex] is [tex]\(\frac{3}{4}\)[/tex].
Now, we use the point-slope form of the equation of a line [tex]\(y - y_1 = m(x - x_1)\)[/tex], where [tex]\((x_1, y_1)\)[/tex] is a point on the line. In this case, the point is [tex]\((-3, 2)\)[/tex].
[tex]\[ y - 2 = \frac{3}{4}(x + 3) \][/tex]
We can simplify this equation:
[tex]\[ y - 2 = \frac{3}{4}x + \frac{3}{4}(3) \][/tex]
[tex]\[ y - 2 = \frac{3}{4}x + \frac{9}{4} \][/tex]
[tex]\[ y = \frac{3}{4}x + \frac{9}{4} + 2 \][/tex]
[tex]\[ y = \frac{3}{4}x + \frac{9}{4} + \frac{8}{4} \][/tex]
[tex]\[ y = \frac{3}{4}x + \frac{17}{4} \][/tex]
Now we convert this back to the standard form [tex]\(Ax + By = C\)[/tex]:
1. Multiply every term by 4 to clear the fractions:
[tex]\[ 4y = 3x + 17 \][/tex]
[tex]\[ 3x - 4y = -17 \][/tex]
So, the simplified equation of the line parallel to [tex]\(3x - 4y = -17\)[/tex] that passes through the point [tex]\((-3, 2)\)[/tex] turns out to be simpler than expected.
Thus, the required line equation is [tex]\(3x - 4y = -17\)[/tex].
However exact result may take is self-reflection based on presumed passed points and following accordance of option may infer refined calculations.
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