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Math  /  Algebra

QuestionUse the given conditions to write an equation for the line in point-slope form and general form. Passing through (−6,1)(-6,1) and parallel to the line whose equation is 5x−2y−3=05 x-2 y-3=0
The equation of the line in point-slope form is □\square (Type an equation. Use integers or fractions for any numbers in the equation.) The equation of the line in general form is □\square =0=0. (Type an expression using xx and yy as the variables. Simplify your answer. Use integers or fractions for

Studdy Solution

STEP 1

1. We need to find the equation of a line passing through the point (−6,1)(-6, 1).
2. The line is parallel to the line given by the equation 5x−2y−3=05x - 2y - 3 = 0.
3. Parallel lines have the same slope.
4. We need to express the equation in both point-slope form and general form.

STEP 2

1. Determine the slope of the given line.
2. Use the slope and the given point to write the equation in point-slope form.
3. Convert the point-slope form equation to general form.

STEP 3

Determine the slope of the given line.
The equation of the given line is 5x−2y−3=05x - 2y - 3 = 0. To find the slope, rewrite it in slope-intercept form y=mx+by = mx + b.
Start by isolating yy: 5x−2y−3=0 5x - 2y - 3 = 0 −2y=−5x+3 -2y = -5x + 3 y=52x−32 y = \frac{5}{2}x - \frac{3}{2}
The slope mm of the given line is 52\frac{5}{2}.

STEP 4

Use the slope and the given point to write the equation in point-slope form.
The point-slope form of a line is given by: y−y1=m(x−x1) y - y_1 = m(x - x_1)
Substitute m=52m = \frac{5}{2}, x1=−6x_1 = -6, and y1=1y_1 = 1: y−1=52(x+6) y - 1 = \frac{5}{2}(x + 6)
The equation of the line in point-slope form is: y−1=52(x+6) y - 1 = \frac{5}{2}(x + 6)

STEP 5

Convert the point-slope form equation to general form.
Start with the point-slope equation: y−1=52(x+6) y - 1 = \frac{5}{2}(x + 6)
Distribute the slope: y−1=52x+15 y - 1 = \frac{5}{2}x + 15
Add 1 to both sides: y=52x+16 y = \frac{5}{2}x + 16
Multiply through by 2 to eliminate the fraction: 2y=5x+32 2y = 5x + 32
Rearrange to get the general form Ax+By+C=0Ax + By + C = 0: 5x−2y+32=0 5x - 2y + 32 = 0
The equation of the line in general form is: 5x−2y+32=0 5x - 2y + 32 = 0

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