Question

# Identify the parallel lines and perpendicular lines from the given set.

2x + y = 1

9x + 3y = 6

y = 3x

y = -3x

2y = 4x +6

Y = - x/2

## The correct answer is: line e (2y = 4x +6) and line f (Y = - x/2) are perpendicular lines & line b (9x + 3y = 6) and line d (y = -3x) are parallel lines

### Hint:-

1. Standard form of equation of a straight line is y = mx + c.

2. Slopes of parallel lines are equal.

3. Slopes of perpendicular lines are negative reciprocals of each other.

Step-by-step solution:-

We will simplify the given equations and compare the same with standard form of a straight line to find the value of m.

a. 2x + y = 1

∴ y = -2x + 1

Comparing the above equation with standard form of a line i.e. y = mx + c, we get- m = -2 ......................... (Equation i)

b. 9x + 3y = 6

∴ 3y = -9x + 6

∴ y = -3x + 2 ............................ (Dividing both sides by 3)

Comparing the above equation with standard form of a line i.e. y = mx + c, we get- m = -3 ......................... (Equation ii)

c. y = 3x

∴ y = 3x + 0

Comparing the above equation with standard form of a line i.e. y = mx + c, we get- m = 3 ......................... (Equation iii)

d. y = -3x

∴ y = -3x + 0

Comparing the above equation with standard form of a line i.e. y = mx + c, we get- m = -3 ......................... (Equation iv)

e. 2y = 4x +6

∴ y = 2x + 3 .................................... (Dividing both sides by 2)

Comparing the above equation with standard form of a line i.e. y = mx + c, we get- m = 2 ......................... (Equation v)

f. Y = - x/2

∴ y = -1/2 x + 0

Comparing the above equation with standard form of a line i.e. y = mx + c, we get- m = -1/2 ......................... (Equation vi)

Final Answer:-

∴ line e (2y = 4x +6) and line f (Y = - x/2) are perpendicular lines & line b (9x + 3y = 6) and line d (y = -3x) are parallel lines.

### Related Questions to study

Carrie, a packaging engineer, is designing a container to hold 12 drinking glasses shaped as regular octagonal prisms. Her initial sketch of the top view of the base of the container is shown above.

Carrie redesigned the container because the initial sketch did not account for cushioning material between the glasses. The area of the base of the newly designed container is greater than the area of the base in the initial sketch. What is the area, in square inches, of the base of the newly designed container?

**Note:**

There is a shorter way of doing this problem. There is a simple rule followed while increasing or decreasing a value by a certain percentage. When we decrease a quantity by x%, we multiply the quantity by (1-0.x). Whereas when we increase a quantity by x%, we multiply it with (1+0.x).

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Carrie redesigned the container because the initial sketch did not account for cushioning material between the glasses. The area of the base of the newly designed container is greater than the area of the base in the initial sketch. What is the area, in square inches, of the base of the newly designed container?

**Note:**

There is a shorter way of doing this problem. There is a simple rule followed while increasing or decreasing a value by a certain percentage. When we decrease a quantity by x%, we multiply the quantity by (1-0.x). Whereas when we increase a quantity by x%, we multiply it with (1+0.x).

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Use the perpendicular line formula to determine whether two given lines are perpendicular. For example, when the slope of two lines is given to compare, we can use the perpendicular line's formula. A 90-degree angle is created by two lines that are perpendicular to one another.

Slope exists on every line. Because it shows how quickly our line is rising or falling, the slope of a line reveals how steep a line is. Mathematically, the slope of a line is known as the ratio of change in the line's y-value to the change in its x-value.

¶A line's slope can be determined using its two points (x_{1}, y_{1}) and (x_{2}, y_{2}). The formula (y_{2} - y_{1}) / is used to find the change in y and divided by the change in x. (x_{2} - x_{1}).

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**Note:**

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If the length and width of the container base in the initial sketch were doubled, at most how many more glasses could the new container hold?

**Note:**

A few simple ideas are used in solving this problem, like, area of a rectangle is given by the product of its length and breadth and the basic idea of division.

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Another answer for this problem could be taken as 137 as well.

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