Question

$14 the siapa-intercept form of the ention of the line that passes through the 2 points(2.3) 16.11)23-2) 6.1)2. (-5.9) 2.0)5. Find an equation of the line that passesThrough the point (37) and has the sameintercept as the line given byY = x-56 Find an equation of the line thatpasses through the point (4-5) andis parallel to the line passing throughthe points (2.0) and (-2,-2)7. Find an equation of the line that passesthrough the point (-4.11) and has the sameY-intercept as the line given byY.5x3.8. Find an equation of the line that passesthrough the point (3.-4) and is parallel to theline passing through the points (-4-1) and(2.5)

Answer 1

2) 3y = -x - 3

x + 3y = -3

3) y = -3x - 6

3x + y = -6

4) y = 3x + 16

-3x + y = 16

6) 2y = x + 6

-x + 2y = 6

Step-by-step explanation

The general form of the equation in point-slope form is

y - y₁ = m (x - x₁)

where

y = y-coordinate of a point on the line.

y₁ = This refers to the y-coordinate of a given point on the line

m = slope of the line.

x = x-coordinate of the point on the line whose y-coordinate is y.

x₁ = x-coordinate of the given point on the line

So, for each of these, we need to find the slope and use the coordinates of one of the given points to write the equation of the line.

For a straight line, the slope of the line can be obtained when the coordinates of two points on the line are known. If the coordinates are (x₁, y₁) and (x₂, y₂), the slope is given as

`Slope=m=\frac{Change\text{ in y}}{Change\text{ in x}}=(y_2-y_1)/(x_2-x_1)`

2) (3, -2) and (-6, 1)

(x₁, y₁) and (x₂, y₂) are (3, -2) and (-6, 1)

`\text{Slope = }(1-(-2))/(-6-3)=(3)/(-9)=-(1)/(3)`

Using (3, -2) as the point (x₁, y₁)

Recall,

y - y₁ = m (x - x₁)

m = -(1/3)

x₁ = 3

y₁ = -2

y - (-2) = -⅓ (x - 3)

y + 2 = -⅓ (x - 3)

Multiply through by 3

3y + 6 = - (x - 3)

3y + 6 = -x + 3

3y + x = -6 + 3

x + 3y = -3

3) (-5, 9) and (-2, 0)

(x₁, y₁) and (x₂, y₂) are (-5, 9) and (-2, 0)

`\text{Slope = }(0-9)/(-2-(-5))=(-9)/(-2+5)=(-9)/(3)=-3`

y - y₁ = m (x - x₁)

m = -3

x₁ = -5

y₁ = 9

y - 9 = -3 (x - (-5))

y - 9 = -3 (x + 5)

y - 9 = -3x - 15

y + 3x = 9 - 15

y + 3x = -6

4) (-6, -2) and (-10, -14)

(x₁, y₁) and (x₂, y₂) are (-6, -2) and (-10, -14)

`\text{Slope = }(-14-(-2))/(-10-(-6))=(-14+2)/(-10+6)=(-12)/(-4)=3`

y - y₁ = m (x - x₁)

m = 3

x₁ = -6

y₁ = -2

y - (-2) = 3 (x - (-6))

y + 2 = 3 (x + 6)

y + 2 = 3x + 18

y - 3x = 18 - 2

y - 3x = 16

6) For this question, we've been given (x₁, y₁) to be (4, -5)

We are then given information to help obtain the slope.

Two parallel lines have the same slopes. So, we can find the slope of the other line and use this for the required line.

(x₁, y₁) and (x₂, y₂) are (2, 0) and (-2, -2)

`\text{Slope = }(-2-0)/(-2-2)=(-2)/(-4)=(1)/(2)`

y - y₁ = m (x - x₁)

m = ½

x₁ = 4

y₁ = -5

y - (-5) = ½ (x - 4)

y + 5 = ½ (x - 4)

Multiply through by 2

2y + 10 = (x - 4)

2y = x - 4 + 10

2y = x + 6

Hope this Helps!!!