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Math  /  Algebra

Question1. Find the equation of the line parallel to y=3x+5y=3 x+5 that passes through the point (2,−4)(2,-4).
2. A line passes through the points (1,2)(1,2) and (3,6)(3,6). Find the equation of a line parallel to this line that passes through the point (0,−3)(0,-3).
3. Determine the equation of a line parallel to y=−23x+7y=-\frac{2}{3} x+7 that passes through the point (4,1)(4,1)
4. Find the equation of a line parallel to y=12x−2y=\frac{1}{2} x-2 that passes through the origin.

Studdy Solution

STEP 1

1. Parallel lines have the same slope.
2. The equation of a line in slope-intercept form is y=mx+b y = mx + b , where m m is the slope and b b is the y-intercept.
3. To find the equation of a line parallel to a given line, we use the same slope as the given line.
4. We will use the point-slope form of a line equation, y−y1=m(x−x1) y - y_1 = m(x - x_1) , to find the equation of the line through a given point (x1,y1)(x_1, y_1).

STEP 2

1. Problem 1: Find the equation of the line parallel to y=3x+5 y = 3x + 5 through the point (2,−4)(2, -4).
2. Problem 2: Find the equation of the line parallel to the line through points (1,2)(1, 2) and (3,6)(3, 6) through the point (0,−3)(0, -3).
3. Problem 3: Find the equation of the line parallel to y=−23x+7 y = -\frac{2}{3}x + 7 through the point (4,1)(4, 1).
4. Problem 4: Find the equation of the line parallel to y=12x−2 y = \frac{1}{2}x - 2 through the origin.

STEP 3

Identify the slope of the given line y=3x+5 y = 3x + 5 . The slope m=3 m = 3 .

STEP 4

Use the point-slope form with the point (2,−4)(2, -4) and slope m=3 m = 3 .
y−(−4)=3(x−2) y - (-4) = 3(x - 2)

STEP 5

Simplify the equation:
y+4=3x−6 y + 4 = 3x - 6 y=3x−6−4 y = 3x - 6 - 4 y=3x−10 y = 3x - 10
The equation of the line is:
y=3x−10 y = 3x - 10

STEP 6

Find the slope of the line through points (1,2)(1, 2) and (3,6)(3, 6).
m=6−23−1=42=2 m = \frac{6 - 2}{3 - 1} = \frac{4}{2} = 2

STEP 7

Use the point-slope form with the point (0,−3)(0, -3) and slope m=2 m = 2 .
y−(−3)=2(x−0) y - (-3) = 2(x - 0)

STEP 8

Simplify the equation:
y+3=2x y + 3 = 2x y=2x−3 y = 2x - 3
The equation of the line is:
y=2x−3 y = 2x - 3

STEP 9

Identify the slope of the given line y=−23x+7 y = -\frac{2}{3}x + 7 . The slope m=−23 m = -\frac{2}{3} .

STEP 10

Use the point-slope form with the point (4,1)(4, 1) and slope m=−23 m = -\frac{2}{3} .
y−1=−23(x−4) y - 1 = -\frac{2}{3}(x - 4)

STEP 11

Simplify the equation:
y−1=−23x+83 y - 1 = -\frac{2}{3}x + \frac{8}{3} y=−23x+83+1 y = -\frac{2}{3}x + \frac{8}{3} + 1 y=−23x+83+33 y = -\frac{2}{3}x + \frac{8}{3} + \frac{3}{3} y=−23x+113 y = -\frac{2}{3}x + \frac{11}{3}
The equation of the line is:
y=−23x+113 y = -\frac{2}{3}x + \frac{11}{3}

STEP 12

Identify the slope of the given line y=12x−2 y = \frac{1}{2}x - 2 . The slope m=12 m = \frac{1}{2} .

STEP 13

Use the point-slope form with the point (0,0)(0, 0) (the origin) and slope m=12 m = \frac{1}{2} .
y−0=12(x−0) y - 0 = \frac{1}{2}(x - 0)

STEP 14

Simplify the equation:
y=12x y = \frac{1}{2}x
The equation of the line is:
y=12x y = \frac{1}{2}x

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