Exams Prep Master
Parallel lines are straight lines that lie on the same plane ( A flat surface that runs up to infinity) and do not cross at any point and these two lines are constantly at the same distance apart or equidistant apart from each other. In other words, two lines will be parallel if they always maintain the same distance apart. Meaning, if two lines are drawn in such a way that they could extend into infinity without ever meeting each other, these lines are referred to as parallel. Parallel lines are also called Equidistant lines. We can also say that the distance between two parallel lines will always be constant. Here, we will learn more about the parallel line formula and discuss some important questions.
| Table of Content |
Key Terms: Parallel Line Definition, Parallel line formula, Slope, Parallel lines, Parallel line Requirements, Parallel Line Equation, Alternating Angles.
Also read: Frequency Polygon
Parallel Lines
[Click Here for Sample Questions]
Parallel lines are are those lines that are created when two lines are drawn in such a way that they may continue to infinity without ever intersecting each other. They are at the same distance apart from each other and have the same slope, so equidistant lines are sometimes also known as parallel lines.
In the above figure, two lines are drawn up to infinity and are equidistant apart from each other, so we can say both the lines i.e, the Red Line and the Blue Line are parallel to each other. They also point in the same direction.
The video below explains this:
Parallel Line Formula Detailed Video Explanation:
Parallel Line Symbol
[Click Here for Sample Questions]
|| is the symbol used to express parallelism between 2 lines.
Example:- Line A || Line B is expressed as Line A is parallel to Line B
Also read:
Parallel Line Requirements
[Click Here for Sample Questions]
If a transversal line cuts two straight lines, and if
- If the two matching angles are identical, then the two straight lines are parallel.
- The two straight lines are parallel to each other if the pair of alternating angles is equal.
- The two straight lines are parallel if the pair of internal angles on the same side of the transversal is supplementary.
Also read: Chance and Probability
Parallel Line Formula
[Click Here for Sample Questions]
From the given equation, we must first determine the slope. Then, using the straight-line equation, determine the value of y.
The slope should be -a/b if the equation of the line is ax + by + c = 0 and the coordinates are (x1, y1).
When two lines are parallel to each other, their slopes are the same.
Consider the slopes of the two parallel lines, m1 and m2. Parallel Lines = m1 = m2
If you want to find out the parallel line of a given line with slope m and which is passed through a point (x1, y1), the formula used is -
Slope = −a/b
Therefore, Parallel Lines Equation is-
y − y1 = m (x − x1)
By comparing the slopes of two lines, we may determine if they are parallel using their equations. The lines are parallel if their slopes are the same but their y-intercepts differ.
The lines are not parallel if the slopes are not same or equal.
Things to Remember
- Two Parallel Lines do not intersect each other at any point.
- They're at equidistant from each other.
- Two Parallel Lines have the same slope.
- f(x) = 2x + 6 and f(x) = 2x - 4 have the same slope and that is 2, hence they're parallel.
Also read:
Sample Question
Ques: Find out the parallel line of the given straight line 6x – 3y = 2 passing through a point (1, 2). (5 marks)
Ans: The given equation is,
6x – 3y = 2 and the coordinates are (1, 2)
Slope = −a/b
m = -a/b = -6 / -3 = 2
(x1, y1) = (1, 2)
Parallel line equation is:
y – y1 = m (x – x1)
y – 2 = 2 (x – 1)
y – 2 = 2x – 2
y = 2x – 2 + 2
y = 2x
Or
2x – y = 0.
Ques: Find the slope of the line parallel to the equation y + 2x = 2. (5 marks)
Ans: Given equation is y + 2x = 2y
It has of two variables x and y
The general equation of a straight line is given by Ax + By + C=0
where A, B, and C are real numbers and also, A and B are non-zero constants.
Therefore, the slope of the straight line given by the equation Ax + By + C = 0 is m = - A / B where B is not equal to zero
Ax+By+C= is −AB
Thus, we have the equation 2x + y − 2 = 0 where the values of A, B, and C are 2, 1, and -2 respectively.
2x+y−2=0
Therefore the slope of the equation 2x + y - 2 = 0 is m = -2/1 = -2
Since we already know that two parallel lines have the same slope, therefore the line parallel to y + 2x = 2 should also have the same slope and that is -2
Hence −2 is the slope of the line parallel to the equation.
Ques: Write the equation of a line that passes through the point (3,1) and is parallel to the line y = 2x + 3. (5 marks)
Ans: Parallel lines have the same slope.
The slope of the line with equation y = 2x+3y is 2 .
So, any line parallel to y = 2x+3y has the same slope 2 .
Now use the Point Slope Form to find the equation.
y−y1 = m(x−x1)
We have to find the equation of the line which has slope 2 and passes through the point (3,1) . So, replace m with 2 , x1 with 3 , and y1 with 1 .
y−1 = 2(x−3)
Use the distributive property
y−1 = 2x−6
Add 1 to each side.
y−1+1 = 2x−6+1
y = 2x−5
Therefore, the line y = 2x−5 is parallel to the line y=2x+3 and passes through the point (3,1).
Ques: Determine whether or not the lines 2y - 4x -10 = 0 and y = 2x + 27 are parallel. (5 marks)
Ans: Parallel lines are to be found.
Given:
Line 1 equation: 2y - 4x -10 = 0
We obtain by rearranging and dividing by two.
2x + 5 = y
When comparing y = m1 x + c1
m1 = 2
we find the slope to be m1 = 2
Line 2's equation is now y = 2x + 27.
When comparing y = m 2 x + c 2
m 2 = 2
Using the parallel line formula we know that the slope of two parallel lines is the same or equal
and since, m1 = m2 = 2
As a result, the lines shown are parallel.
Ques: Find the slope of a line that is parallel to the line 12x + y + 90 = 0. (5 marks)
Ans: Slope of a line parallel to 12x + y + 90 = 0 will be
Given:
12x + y + 90 = 0 is the equation for the given line.
When we rearrange the numbers, we obtain y = -12x - 90.
When we compare y = m 1 x + c 1,
we obtain slope m = -12.
For lines to be parallel, their slopes must be equal, according to the parallel line formula.
As a result, any line parallel to this line must have a slope of m = -12.
As a result, the line parallel to the supplied line has a slope of -12.
Ques: There is a line defined by the equation below: 3x+4y=12, There is a second line that passes through the point (1,2) and is parallel to the line given above. What is the equation of this second line? (5 marks)
Ans: The slope of parallel lines is the same. Convert the equation to slope-intercept form to find the slope in the first line.
12 = 3x + 4y
–3x + 12 = 4y
y = –(3/4)x + 3
Slope m = –3/4
We know the slope of the second line will be –3/4, and we are given the point (1,2). We can write an equation in slope-intercept form and solve for the y-intercept using these numbers.
y = mx + b
2 = –3/4(1) + b
2 = –3/4 + b
b = 2 + 3/4 = 2.75
Putting the value y-intercept back into the equation to get our final answer.
y = –(3/4)x + 2.75
Ques: What is the equation of a line that is parallel to the line y= (1/2)x + 3 and includes the point (4, 2)? (2 marks)
Ans: The line parallel to y=1/2x + 3 must have a slope of 1/2, giving us the equation y=1/2x+b. To solve for b, we can substitute the values for y and x.
2= (1/2)4 +b
2=2+b
b=0
Therefore, the equation of the line is y=1/2x
Ques: Which of the following answer choices gives the equation of a line parallel to the line: y = -2x +4: (2 marks)
y = -2x +4
y = 2x
y = -2x -1
y = 2x +4
Ans: Since we know that those lines that have the same slope but different y-intercepts are parallel lines. When the equations of two lines are the same they have infinitely many points in common, whereas parallel lines have no points in common.
Our equation is given in slope-intercept form,
y=mx+b
where m is the slope. In this particular situation m=-2.
Therefore we want to find an equation that has the same m value and a different b value.
Thus,
y = -2x -1 is parallel to our equation
Mathematics related links:
Comments