The following Topics and Sub-Topics are covered in this chapter and are available on MSVgo:
One of the most important sections of the class 10th board exams is the topic ‘Linear Equations.’ Here you will learn to use two or more variables like x, y, z, along with the constants to define any function. Do you remember the definition of functions and equations? Read on!
In this method, we are given a pair of linear equations for which we need to find the solutions. The best way is to turn both the equations into dependent function forms, and then plot their graphs.
Example: You are given two linear equations, namely
Now both these equations can be written as:
Now simply, mark all the points of x and y on the graph and form two lines. The point where the lines intersect each other is the solution to the given question.
In this method, we are given a pair of linear equations for which we need to find the solutions. We convert both the equations into the dependant function form by bringing one variable to one side of the = sign. And then we compare them. The solutions can be found in four possible ways, as depicted below.
Example: You are given two linear equations, namely
Now both these equations can be written as
Now, you take the value of x from the first equation and put it in the second equation in the place of x. Then after solving it, you get the value of y. Put that value in any one of the equations to get the value of x.
Example: You are given two linear equations, namely
You now add or subtract both the equations in such a way that any one variable gets cancelled. In the above example, you can simply add both equations, which would result in:
-4x = 8
-x = 2
Now put the value of x in any equation to get the value of y.
This method follows a simple formula, which is given as
You have two linear equations
You can calculate x and y with this formula.
All the above cases were for the linear equations, where the two variables can easily be checked. For some other examples, like, 2/x + 3/y = 5, we can simply assume 1/x = u and 1/y= w and change the given equation into something like this: 2u + 3w = 5. This is particularly for equations Reducible To A Pair Of Linear Equations In Two Variables.
After we have calculated u and w, we can easily put the real equations in place of u and w, then find the values of x and y.
Linear equation in two variables is an easy topic that is asked in the class 10th boards exam. There are various ways of solving a pair of linear equations, such as substitution method, graphical method, cross-multiplication method, and more. Choose one which is suitable for the question and solves it in less time.
A linear equation is an equation in which all the operations between variables and constants are linear. For example,
a.x + b.y + c = d
+ and – are linear operations; they do not span dimensions and work on the same dimension.
* and / are different.
If the lines turn out to be ax + by = c and ax – by = d or ax + by = c and dx (+/-) ey = f, then you have two solutions. If you get any value as zero, it is still a well-defined solution.
If both lines are ax + by = c, then you have one solution, that is the whole line. You will not get a well defined solution.
If lines are ax + by = c and ax + by = d, then you have no solutions, as these two lines are parallel. Distance between the two lines will be |c-d| / √(a^2+b^2).
There are many possibilities for solving two variable equations.
An equation is a certain kind of relation between an indeterminate and one or two constants (with respect to that indeterminate). By custom, the indeterminate is often symbolised as x or a, and either numeric constants or other alphabetic characters are used for the constants such as x, y, z, or a, b, c, and so on. It’s a relation that involves equality. In its simplest form, we have something like: ax + by = c
Some more examples are given below:
The graph is an intersection between the two poles at 90 degrees. On one side, you can show the dependent function (y), and on the other side, there is the independent argument (x).
You can get more such questions for practice on MSVgo. There are some formulas of Pair Of Linear Equations In Two Variables that are critical and you can check out videos on MSVgo to understand the concept behind them.