Math Examples of Linear Equations

Lines are the equations that pass through many Collinear Points or we can also say that points lying on a line satisfy its equation. The Point – slope form of a line can be written using any one point lying on the line. For instance, suppose we have a point (a, b) that lies on the line and Slope of the line is 's', then equation of line in point – slope form can be written as follows:
Y – a = s (X – b)............. equation 1

Slope intercept form of line is given as:
Y = sX + c........... equation 2
Where, 'c' is the intercept made by the line:

When we have two points lying on the line, we can calculate the value of slope by formula:
s = (b₂ – b₁) / (a₂ – a₁).......... equation 3

Let us consider an example to understand this concept:
Example: How do I find the slope intercept and point form of the equation (-1, 7) and (12, 5)?
Solution: Two points are: (-1, 7) and (12, 5). Using these points we can calculate the slope of equation using the formula shown in equation 3 as follows:
Slope = s = (b₂ – b₁) / (a₂ – a₁),
Or s = (5 – 7) / (12 + 1) = -2 / 13,
Substituting the value of 's' in the equation 2, we can write:
Y = -2 / 13 * X + C........... equation 4
Substituting (1, 7) in equation 4 we get:
7 = -2 / 13 * 1 + C,
Or C = 7 + 2/13 = 93/13,
So, the point form and slope – intercept form of line can be given as:
Y - 7 = -2/13 (X – 1),
Or Y = -2/13 + 93/13.

For finding ‘x’ and ‘y’ intercepts we need to follow the below steps
Step 1: In the first step we find x – intercepts, and Set y = 0 and solve for ‘x’,
We write the given equation x + y = 1,
Here we assume y = 0,
x + (0) = 1,
x = 1,  
By this step we get x = 1,
Step 2: in this step we find y- intercepts and set x =0 and solve for ‘y’
Here we put x =0 in given equation,
       0 + y = 1,
        y = 1,
Now we get y = 1,
Step 3: This is last step of the problem in this step we write ‘x’ and ‘y’ intercepts value
For x = 1, so x- Intercepts is (1, 0).
For y =1, so y- intercepts is (0, 1).

For finding ‘x’ and ‘y’ intercepts we need to follow the below steps
Step 1: In the first step we finding x– intercepts, and Set y = 0 and solve for ‘x’
 We write the given equation y = x² + 2x +1,
Here we assume y = 0,
x² + 2x + 1 = 0,
x² + x +x +1 = 0,
 x (x + 1) + 1(x + 1) = 0,
(x + 1) (x + 1) = 0,
By this step we get x = -1,-1
Step 2: in this step we find y- intercepts and set x =0 and solve for ‘y’
Here we put x =0 in given equation,
y = 1,
Now we get y = 1,
Step 3: This is last step of the problem in this step we write ‘x’ and ‘y’ intercepts value
For x = -1, -1, so x- Intercepts is (-1, 0), (-1, 0),
For y =1, so, y- intercepts is (0, 1).

For finding ‘x’ and ‘y’ intercepts we need to follow the below steps
Step 1: In the first step we find x–intercepts, and Set y = 0 and solve for ‘x’
 We write the given equation 9x + 4y = 36,
Here we assume y = 0,
9x + 4(0) = 36,
x = 36/9 = 4,
By this step we get x = 4.
Step 2: in this step we find y- intercepts and set x =0 and solve for ‘y’,
Here we put x =0 in given equation,
9(0) + 4y = 36,
        Y = 9,
Now we get y =9
Step 3: This is last step of the problem in this step we write X and Y intercepts value
For x = 4, so x- Intercepts is (4, 0),
For y =9, so y- intercepts is (0, 9).

For finding ‘x’ and ‘y’ intercepts we need to follow the below steps
Step 1: In the first step we find x–intercepts, and Set y = 0 and solve for ‘x’,
 We write the given equation 4x + 3y = 16
Here we assume that the value of y = 0
4x + 3(0) = 16
x = 16/4 = 4
By this step we get value of x = 4
Step 2: In this step we find y- intercepts and set x =0 and solve for ‘y’
Here we put x =0 in given equation
4(0) + 3y = 16,
        Y = 16/3,
By this step we get value of y = 16/3,
Step 3: This is last step of the problem in this step we write ‘x’ and ‘y’ intercepts value
For x = 4, so x- Intercepts is (4, 0),
For y =16/3, so y- intercepts is (0, 16/3).

As we know that the formula for the Point slope form is,
(y – b) = m (x - a).
In our question the value of ‘a’ is 0 and value of ‘b’ is 0 and value of ‘m’ is 3, so we will put all these values in our equation to get the exact solution.
y –  0 = 3(x - 0),
y = 3x,
y - 3x = 0,
This is the required point Slope form.
Now for getting the standard form we need the value of coefficient of ‘x’ will be greater than zero, but in our point Slope form it is less than zero , so we will multiply by the minus sign in whole equation to get the standard equation,
-y + 3x = 0,
3x – y = 0,
Here the value of ‘A’ is 3, the value of ‘B’ is -1 and value of ‘C’ is 0.
This is the required solution for the question.

As we know that the formula for the Point slope form is,
(y – b) = m (x - a).
In our question the value of ‘a’ is 2 and value of ‘b’ is four and value of ‘m’ is 3, so we will put all these values in our equation to get the exact solution.
y – 4 = 3(x - 2),
y - 4 = 3x – 6,
y - 3x = -2,
This is the required point Slope form.
Now for getting the standard form we need the value of coefficient of ‘x’ will be greater than zero, but in our point Slope form it is less than zero, so we will multiply by the minus sign in whole equation to get the standard equation,
-y + 3x = 2,
3x – y = 2,
Here the value of ‘A’ is 3, the value of ‘B’ is -1 and value of ‘C’ is 2.
This is the required solution for the question.

As we know that the formula for the Point slope form is,
(y – b) = m (x - a).
In our question the value of ‘a’ is 0 and value of ‘b’ is four and value of ‘m’ is 5, so we will put all these values in our equation to get the exact solution.
y – 4 = 3(x - 0),
y - 4 = 3x,
y - 3x = 4,
This is the required point Slope form.
Now for getting the standard form we need the value of coefficient of ‘x’ will be greater than zero, but in our point Slope form it is less than zero, so we will multiply by the minus sign in whole equation to get the standard equation,
-y + 3x = -4,
3x – y = -4,
Here the value of ‘A’ is 3, the value of ‘B’ is -1 and value of ‘C’ is -4.
This is the required solution for the question.

As we know that the formula for the Point slope form is,
(y – b) = m (x - a).
In our question the value of ‘a’ is 1 and value of ‘b’ is four and value of ‘m’ is 3, so we will put all these values in our equation to get the exact solution.
y – 4 = 3(x - 1),
y - 4 = 3x – 3,
y - 3x = 1,
This is the required point Slope form.
Now for getting the standard form we need the value of coefficient of ‘x’ will be greater than zero, but in our point Slope form it is less than zero, so we will multiply by the minus sign in whole equation to get the standard equation,
-y + 3x = -1,
3x – y = -1,
Here the value of ‘A’ is 3, the value of ‘B’ is -1 and value of ‘C’ is -1. 
This is the required solution for the question.

Whenever we see this type of problem firstly we need to have good knowledge of Point slope form and standard form, firstly we will discuss point Slope form then we will move to standard form. If we are having two points as (a, b) and ‘x’ and ‘y’ as variables and ‘m’ as the Slope of the line, then we can find the equation of line by the formula given below,
(Y – b) = m (x - a).
If the line is passing through the point (a, b) and (c, d) and slope of the line is not given then we can find the slope of the line as
m = d – b / c – a,
Now moving to standard form, if we are having two variable ‘x’ and ‘y’ and A, B and C as constant then Ax + By = C is the standard form of the equation if A, B and C are Integer and A > 0 and ‘B’ is not equal to zero.
Now moving to our question point slope form will be,
Y – 4 = 3 (x - 2),
ð Y – 4 = 3x -6,
ð Y -3x = -6 + 4,
ð Y -3x = -2,
This is the required point slope form.
Now for getting the standard form we need the value of coefficient of x will be greater than zero, but in our point slope form it is less than zero , so we will multiply by the minus sign in whole equation to get the standard equation,
-y + 3x = 2,
We can also write this equation as,
3x – y = 2,
Here the value of ‘A’ is 3, value of ‘B’ is -1 and ‘C’ is 2.

The solution of above example is as follows:
Here we can easily that we have two co -ordinates (1, 4), (3, 6) and we have to find out Slope.
As we know if we have two co-ordinates (x₁, y₁) and (x₂, y₂) then the Slope
m = (y₂ – y₁)/ (x₂ – x₁),
So the slope ‘m’
m = (6 – 4) / (3 – 4) = 2 / -1 = -2,
The slope between two co- ordinates is -2.

The solution of above example is as follows:
Here we can easily see that we have an equation of straight line 3x – y = 3 which is in standard form Ax + By + c = 0.
To find the Slope we have to convert that in Slope equation which is
y = mx + c,
Where ‘m’ is slope of line.
So 3x – y = 3,
Now we move 3x term right hand side
-y = -3x + 3,
y = 3x – 3,
So the slope of line is m = 3.

The solution of above example is as follows:
Here we can easily that we have two co -ordinates (-4, 5), (-5, 4) and we have to find out Slope.
As we know if we have two co-ordinates (x₁, y₁) and (x₂, y₂) then the Slope,
m = (y₂ – y₁)/ (x₂ – x₁),
So the slope ‘m’
m = (4 - 5) / (-5 + 4) = -1 / -1 = 1,
The slope between two co -ordinates is 1.

The solution of above example is as follows:
Here we can easily that we have two co -ordinates (-5, 6), (-7, 3) and we have to find out Slope.
As we know if we have two co-ordinates (x₁, y₁) and (x₂, y₂) then the Slope
m = (y₂ – y₁)/ (x₂ – x₁),
So the slope ‘m’
m = (3 - 6) / (-7 + 5) = -3 / -2 = 3/ 2,
The slope between two co -ordinates is 3/ 2.

The solution of above example is as follows:
Here we can easily that we have two co -ordinates (-2, 3), (-5, 4) and we have to find out Slope.
As we know if we have two co-ordinates (x₁, y₁) and (x₂, y₂) then the Slope
m = (y₂ – y₁)/ (x₂ – x₁),
So the slope ‘m’
m = (4 - 3) / (-5 + 2) = -1 / 3 = -1/ 3,
The slope between two co -ordinates is -1/ 3.