DIFFERENT FORMS OF EQUATION OF STRAIGHT LINE

Suppose that we have the graph of a straight line and we want to find its equation.

For any straight line, if we want to find the equation, we must have the following information of that straight line. 

1. Slope and y-intercept

2. One point and slope

3. Two points

4. x-intercept and y-intercept

If we have any one of the five information above,  we can find the equation of a line using the formulas given below. 

1. Slope-Intercept form equation of a line :

y = mx + b

m ---> slope

b ----> y-intercept

2. Point-Slope form equation of a line :

y - y₁ = m(x - x₁)

m ---> slope

(x₁, y₁) ----> point

3. Two-Points form equation of a line :

(y - y₁)/(y₂ - y₁) = (x - x₁)/(x₂ - x₁)

Two points ----> (x₁, y₁) and (x₂, y₂)

4. Intercept-Form equation of a line :

x/a + y/b = 1

a ----> x-intercept

b ----> y-intercept

Apart from the above forms of equation of straight line, there are some other ways to get equation of a straight line.

1. If a straight line is passing through a point (0, k) on y-axis and parallel to x-axis, then  the equation of the straight line is y = k.

2. If a straight line is passing through a point (c, 0) on x-axis and parallel to y-axis, then  the equation of the straight line is x = c.

3. Equation of x-axis :

y = 0

(Because, the value of y in all the points  on x-axis is zero)

4. Equation of y-axis :

x = 0

(Because, the value of x in all the points  on y-axis is zero)

5. General equation of a straight line :

ax + by + c = 0

Solved Problems

Problem 1 :

Find the equations of the straight lines parallel to the coordinate axes and passing through the point (3, -4).

Solution :

Let L and L' be the straight lines passing through the point (3, -4) and parallel to x-axis and y-axis respectively.

The y-coordinate of every point on the line L is – 4.

Hence, the equation of the line L is

y = -4

Similarly, the x-coordinate of every point on the straight line L' is 3.

So, the equation of the line L' is

x = 3

Problem 2 :

Find the general equation of the straight line whose angle of inclination is 45° and y-intercept is 2/5.

Solution :

From the angle of inclination 45°, we can get the slope.

Slope of the line :

m = tan45°

m = 1

Given : y-intercept b = 2/5.

Equation of a straight line in slope-intercept form :

y = mx + b

Substitute m = 1 and b = 2/5.

y = x + 2/5

Multiply each side by 5.

5y = 5x + 2

Subtract 5y from each side.

0 = 5x - 5y + 2

5x - 5y + 2 = 0

Problem 3 :

Find the general equation of the straight line passing through the point (-2, 3) with slope 1/3.

Solution :

Given : Point = (-2, 3) and slope m = 1/3.

Equation of a straight line in point-slope form :

y - y₁ = m(x - x₁)

Substitute (x₁ , y₁) = (-2 , 3) and m = 1/3.

y - 3 = (1/3)(x + 2)

Multiply each side by 3.

3(y - 3) = x + 2

Simplify.

3y - 9 = x + 2

Subtract 3y from each side. 

-9 = x - 3y + 2

Add 9 to each side.

0 = x - 3y + 11

x - 3y + 11 = 0

Problem 4 :

Find the general equation of the straight line passing through the points (-1, 1) and (2, -4).

Solution :

Given : Two points on the straight line : (-1, 1) and (2, -4).

Equation of a straight line in two-points form :

(y - y₁)/(y₂ - y₁) = (x - x₁)/(x₂ - x₁)

Substitute (x₁ , y₁) = (-1, 1) and (x₂, y₂) = (2, -4).

(y - 1)/(-4 - 1) = (x + 1)/(2 + 1)

(y - 1)/(-5) = (x + 1)/3

3(y - 1) = -5(x + 1)

3y - 3 = -5x - 5

5x + 3y + 2 = 0

Problem 5 :

The vertices of a triangle ABC are A(2, 1), B(-2, 3) and C(4, 5). Find the equation of the median through the vertex A.

Solution :

Median is a straight line joining a vertex and the midpoint of the opposite side.

In ΔABC above, midpoint of BC :

= D((-2 + 4)/2, (3 + 5)/2)

= D(1, 4)

The median through A is the line joining two points A (2, 1) and D(1, 4).

Equation of the median through A :

(y - y₁)/(y₂ - y₁) = (x - x₁)/(x₂ - x₁)

Substitute (x₁ , y₁) = (2, 1) and (x₂, y₂) = (1, 4). 

(y - 1)/(4 - 1) = (x - 2)/(1 - 2)

(y - 1)/3 = (x - 2)/(-1)

-1(y - 1) = 3(x - 1)

-y + 1 = 3x - 3

3x + y - 4 = 0

Problem 6 :

If the x-intercept and y-intercept of a straight line are 2/3 and 3/4 respectively, find the general equation of the straight line.

Solution :

Given : x- intercept is 2/3 and y-intercept is 3/4.

Equation of a straight line in intercept form :

x/a  +  y/b = 1

Substitute a = 2/3 and b = 3/4.

x/(2/3) + y/(3/4) = 1

3x/2 + 4y/3 = 1

(9x + 8y)/6 = 1

Multiply each side by 6.

9x + 8y = 6

Subtract 6 from each side from 6.

9x + 8y - 6 = 0

Problem 7 :

Find the equations of the straight lines each passing through the point (6, -2) and whose sum of the intercepts is 5.

Solution :

Let 'a' and 'b' be the x-intercept and y-intercept of the required straight line respectively.

Given : Sum of the intercepts is 5.

a + b = 5

Subtract 'a' from each side.

b = 5 - a

Equation of a straight line in intercept form :

x/a + y/b = 1

Substitute b = 5 - a.

x/a + y/(5 - a) = 1

[(5 - a)x + ay]/a(5-a) = 1

(5 - a)x + ay = a(5 - a) ----(1)

The line is passing through (6, -2).

So, substitute (x, y) = (6, -2).

(5 - a)6 - 2a = a(5 - a)

30 - 6a - 2a = 5a - a²

a² - 13a + 30 = 0

a² - 13a + 30 = 0

(a - 10)(a - 3) = 0

a = 10 and a = 3

When a = 10,

 (1)----> (5 - 10)x + 10y = 10(5 - 10)

-5x + 10y = -50

5x - 10y  - 50 = 0

x - 2y - 10 = 0

When a = 3,

(1)---->(5 - 3)x + 3y = 3(5 - 3)

2x + 3y = 6

Problem 8 :

Find the general equations of the straight lines parallel to x- axis which are at a distance of 5 units from the x-axis.

Solution :

From the given information, we can sketch the two lines as shown below. 

One line is above the x-axis at a distance of 5 units. And another line is below the x-axis at a distance of 5 units.

So, y = 5 and y = -5 are the required straight lines.

Problem 9 :

Find the slope and y-intercept of the straight line whose equation is 4x - 2y + 1 = 0.

Solution :

Since we want to find the slope and y-intercept, let us write the given equation 4x - 2y + 1 = 0 in slope-intercept form.

4x - 2y + 1 = 0

4x + 1 = 2y

(4x + 1)/2 = y

2x + 1/2 = y

y = 2x + 1/2

The above form is in slope intercept form.

Comparing y = 2x + 1/2 and y = mx + b,

m = 2 and b = 1/2

Problem 10 :

A straight line has the slope 5. If the line cuts y-axis at -2, find the general equation of the straight line.

Solution :

Since the line cuts y-axis at -2, clearly y-intercept is -2.

So, the slope m = 5 and y-intercept b = -2.

Equation of a straight line in slope-intercept form :

y = mx + b

Substitute m = 5 and b = -2.

y = 5x - 2

Subtract y from each side. 

5x - y - 2 = 0

Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.

WORD PROBLEMS

HCF and LCM  word problems

Word problems on simple equations 

Word problems on linear equations 

Word problems on quadratic equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation 

Word problems on unit price

Word problems on unit rate 

Word problems on comparing rates

Converting customary units word problems 

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles 

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems 

Profit and loss word problems 

Markup and markdown word problems 

Decimal word problems

Word problems on fractions

Word problems on mixed fractions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and Venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed 

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS 

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6