Linear equations are used to represent straight lines on a coordinate plane. A linear equation in two variables can be written in the form
ax + by + c = 0, where a, b, and c are real numbers, and a and b are not both zero.
The solutions of a linear equation are ordered pairs (x, y) that satisfy the equation. When these solutions are plotted on the coordinate plane, they form a straight line. For special cases, when y is constant, the line is parallel to the x-axis, and when x is constant, the line is parallel to the y-axis.
The equations of such lines can be written as: y - k = 0 or x - k = 0 ,where k is any real constant.
The equation x=k represents all points whose x-coordinate is k, while the value of y can be any real number. Therefore, the graph of x=k is a vertical line parallel to the y-axis.
For example, the equation x−2 = 0 can be written as:x + 0y − 2=0
Here, the value of x must be 2, while y can take any value. Hence, all points of the form (2,y) lie on this line, forming a line parallel to the y-axis.
| x | 2 | 2 | 2 | 2 |
| y | -5 | -3 | 3 | 10 |
Similar analysis can be done for the equation y - 2 = 0 where values of x don't affect the equation thus all the values of x can be included in the equation. Thus,
| x | -5 | -3 | 3 | 5 |
| y | 2 | 2 | 2 | 2 |
Equations Passing through Origin
Some equations have (0, 0) as a solution. The plot for such equations will always pass through the origin.
For example: y = 2x
(0,0) satisfies these equations, which means it is one of the solutions and the graph of this equation now must pass through the origin. Let's plot its graph by finding out other solutions.
| x | 0 | 1 | 2 | 3 |
| y | 0 | 2 | 4 | 6 |
These equations are of the form y = cx, where c is a real constant. Since substituting x = 0 gives y = 0, the point (0, 0) always satisfies the equation.
Sample Problems
Question 1: Give geometric representation of 5x + 2y = 10.
Let's find out the solutions to this equation. Assume values for one variable, reduce the equation to a single variable form and then find out the value for other variable.
x 0 1 2 ... y 5 2.5 0 ....
Question 2: Give geometric representation for x = 10.
As explained in the previous sections, this can be represented as, x + 0.y = 10
Now for x = 10 and any other value of "y" it will work. So, the solutions are (10,y) where y can be anything.
Question 3: Plot the graph for the equation 4x = 2y.
Let's find out the solutions to this equation. Assume values for one variable, reduce the equation to a single variable form and then find out the value for other variable.
x 0 1 2 ... y 0 2 4 ....
Question 4: We have an equation 3x + ay = 12. It is known that (2, 3) is a solution to this equation. Find out the value of a.
The solution the equation should satisfy it. (2,3) is a solution.
3(2) + a(3) = 12
⇒ 6 + 3a = 12
⇒ 3a = 6
⇒ a = 2.
Question 5: We know that the force applied to the body is directly proportional to its acceleration. The proportional is given by the mass of the body. Let's say have a body of 5Kg. Plot the Force and acceleration curve for this body.
It's given that force applied to the body is directly proportional to its acceleration. Let's say "F" is the force applied and "a" is the acceleration. F = ka
Here, "k" is the constant of proportionality which is given by the mass of the body.
So, the equation becomes : F = 5a
Now let's plot the solutions to this equation,
a 0 1 2 3 F 0 5 10 15
Question 6: Give the geometric representation of y = 5 as
- One variable equation
- Two variable equation
1. One Variable equation.
y = 5 has only one solution, y = 5. It can be represented on number line
2. Two Variable Equation
y = 5 can be represented as 0.x + y = 5.
Plotting the solutions to this,
x 0 2 -10 y 5 5 5
