A Linear indicates the straight line. We need to find out the linear relation between the two variables. Here, the two variables are x and y. We need to draw the graph for the linear relation between x and y. The basic expression for the linear relation with two variables in mathematics is ax + by + c = 0. Here, a, b, c are constants, and x and y are variables. The constants or real numbers are a and b. These real numbers are not equal to zero and that is called a linear equation with two variables. The below diagram is the basic diagram of the Graph of Standard Linear Relations between x and y.

## Quadrants of a Graph

Generally, the graph is divided into four quadrants. In the first quadrant both the variables, x and y are positive numbers. Where the x variable values are negative and the y variable values are positive, that is called the second quadrant of the graph. In the third quadrant, both the variables x and y are negative numbers. Finally, the fourth quadrant has the y variables as negative numbers and the x variables as positive numbers. That is

- The first quadrant – x and y variables are positive.
- The second quadrant – x variable is negative and the y variable is positive.
- The third quadrant – x and y variables are negative.
- The fourth quadrant – x variable is positive and y variable is negative.

### Linear Relationship with Two Variables

In this section, we are elaborating on how we can find the variable values by using the linear equation.

x value | x + 4 = y | y value | (x, y) |
---|---|---|---|

2 | 2 + 4 = y | 6 | (2, 6) |

3 | 3 + 4 = y | 7 | (3, 7) |

0 | 0 + 4 = y | 4 | (0, 4) |

-1 | -1 + 4 =y | 3 | (-1, 3) |

-2 | -2 + 4 = y | 2 | (-2, 2) |

### Plotting Points on a Graph with x and y Values.

(I) If x = 0 and y = 1, 2, -1, -2.

The given details are x = 0 and y = 1, 2, -1, -2.

The x and y variable values are mentioned on the below graph.

Here, x=0. So, all points are placed on the y- axis.

(ii) If y = 0 and x = 1, 2, -3, -6.

As per the given information y = 0 and x = 1, 2, -3, -6.

The x and y values are marked on the below graph.

All the x and y related points are marked on the x-axis only.

(iii) If x = y

The x and y variables are marked on the below graph.

### Solved Examples on Linear Relationship between x and y

1. If x = 1, 2, 0, -1, -2 and the linear equation is 2x + y +1 =0, then find the y values?

Solution:**
**As per the given information, x = 1, 2, 0, -1, -2.

The given linear equation is 2x + y + 1 = 0.

Substitute the ‘x’ values in the above linear equation, then we will get

If x = 1

Substitute x = 1 in the linear equation 2x + y + 1 = 0.

That is, 2(1) + y + 1 = 0.

3 + y = 0.

So, y = -3.

If x = 1 then y = -3.

If x = 2.

Substitute x = 2 in the linear equation 2x + y + 1 = 0.

That is, 2(2) + y + 1 = 0.

4 + y + 1 = 0.

5 + y = 0.

So, y = -5.

If x = 2 then y = -5.

If x = 0.

Substitute x = 0 in the linear equation 2x + y + 1 = 0.

That is, 2(0) + y + 1 = 0.

y + 1 = 0.

So, y = -1.

If x = 0 then y = -1.

If x = -1.

Substitute x = -1 in the linear equation 2x + y + 1 = 0.

That is, 2(-1) + y + 1 = 0.

– 2 + y + 1 = 0.

-1 + y = 0.

So, y = 1.

So, If x = -1 then y = 1.

If x = -2.

Substitute x = -2 in the linear equation 2x + y + 1 = 0.

That is, 2(-2) + y + 1 = 0.

-4 + y + 1 = 0.

-3 + y = 0.

So, y = 3.

So, If x = -2 then y = 3.

Finally, x and y variable values are

2. Find the two variable values by using the linear equation 2x + 3y = 10?

Solution:

From the given details, the linear equation is 2x + 3y = 10.

To find out the variable values, we need to substitute the x =0 and y = 0 in the linear equation.

Firstly, apply the x = 0 in the above linear equation. That is, 2x + 3y =10.

2(0) + 3y = 10.

3y = 10.

y = 10 / 3.

Now, apply y = 0 in the linear equation. Then we will get 2x + 3y = 10.

2x + 3(0) = 10.

2x = 10.

x = 10 / 2 = 5.

Therefore, the values of the variables are x = 5 and y = 10 / 3.

3. Find the variable values by using the linear equations 2x + 5y =10 and 3x + 6y = 6?

Solution:

The given linear equations are

2x + 5y = 10——–(1).

3x + 6y = 6———(2).

Multiply the equation (1) with 3 on both sides. That is,

3 * (2x + 5y) = 10 * 3.

6x + 15y = 30——–(3).

Multiply equation (2) with 2 on both sides. That is,

2 * (3x + 6y) = 6 *2.

6x + 12y = 12——–(4).

Subtract the equation (3) and equation (4). That is

6x + 15y = 30.

6x + 12y = 12.

(-) (-) (-)

3y = 18.

Y = 18 / 3 = 6.

Substitute the y = 6 in the equation (1). We will get

2x + 5y = 10.

2x + 5(6) = 10.

2x + 30 = 10.

2x = 10 – 30 = -20.

X = -20 / 2 = -10.

Therefore, the two variable values are x = -10 and y = 6.