Reducing equations is the process of simplifying them. Since not all equations are presented in a linear form, it is vital to reduce them in simpler forms that are easy to understand. Performing it includes various mathematical approaches, and they vary as per the need of a particular equation.
The primary aim of this process is to ease the calculation process. Simplifying such complicated equations into their linear form means effortless calculation, and lesser mistakes. In this chapter of reducing method class 10 students will come across various examples of such equations, and the process to abridge them to perform a hassle-free calculation.
Before moving on to the simplification process, one needs to learn more about linear equations. These equations are known as the first order. Also, these are known as equations for a straight line.
A linear equation is an algebraic expression where every term is a product of single variables and constants, or they remain constant themselves. It carries the first-order power of variables. Such equations are typically represented as Xa+Y=0, where X and Y are constants and X is not equal to 0.
Reducing method class 10 teaches students to learn converting different complicated equations into linear or simpler form.
Here are different calculation methods of the same:
Some of them have one variable on the left-hand side, like 4x + 5 = 30.
Some may have one variable, but on both sides, like 4x + 5 = 20 + 6x.
Apart from these, some linear equations may have non-linear forms. It requires reducing equations to linear form to make the calculation process simple. An example here is (2y + 5)/(4y + 2) = 1/4. Such equations are not easy to solve; it requires simplification.
Reducing Equations is the process that converts non-linear ones into linear ones. Since every equation is not always available in a simple and straightforward format, it is essential to break them down to make solving easy.
Solving these equations requires usage of some mathematical applications such as cross multiplication, division, etc. on both sides. It helps to convert complicated equations to their linear forms. Following this conversion, it becomes easy to find the value of the variables.
Among many tactics to simplify non-linear equations, cross multiplication is one of the prominent ones. In this method, students can multiply the numerator of one fraction with another’s denominator, and vice-versa.
Cross multiplication of equation reducible to linear form example includes the following:
(a - 2)/(a + 8) = ⅔
Now, cross multiplying this equation will result in, 3(a - 2) = 2(a + 8).
Following this cross multiplication, one needs to implement another mathematical operation to move a step closer to solve this equation. It is known as opening the brackets. Additionally, another law used here is called distributive law. Under this, students need to multiply any value within the brackets with the one outside of it.
Now, on using this law on the above mentioned equation, one will get:
3a - 6 = 2a + 16
After implementing distributive law, one needs to arrange the variables on one side and constants on one side. While performing this step, students need to remember that, when they move any value from the RHS to the LHS, it will shift from its negative value to a positive one, and vice-versa. Implementing that in this equation results in:
3a - 2a = 16 + 6
a = 22
A point to note here is that, if students do addition, or subtract, or even perform multiplication with the same value on either side, they will get the value of a variable without changing the final equation.
Now, this is a relatively simple example of the concept of reducing method class 10. There are more complex examples as well, where students need to employ more mathematical applications like LCM to find the desired result.
Point to Note: Equations are a condition of a particular variable.
Reducing method class 10 is an essential chapter of mathematics, and helps students get a clear idea of solving equations. Since it is a vital chapter for the upcoming board exams as well as for higher studies, one must learn it in detail, and thoroughly.
Along with the traditional textbooks, and practice sets, online platforms like Vedantu can be a big help for students. The availability of exam notes, mock question papers, study material coupled with live online classes, and doubt clearing sessions let individuals better their exam preparations.
1. What is LCM?
Ans. LCM stands for least common multiple, and it is the smallest multiple, which are common between two numbers. It represents a value that can evenly divide any two numbers is derived from. It is also known as the least common divisor. For instance, the LCM of 4 and 6 is 24, and this 24 is divisible by both 4, 6.
LCM has different applications based on an equation that you are solving. You can use it to add or subtract any two fractions when their denominators are different. During any mathematical operations such as addition, subtraction, mainly with fractions, LCM is used to make the denominators similar. It simplifies the entire process and helps with the calculation.
2. What is HCF?
Ans. HCF stands for the highest common factor, and it represents the greatest number that divides two numbers. It is also termed as Greatest Common Divisor and Greatest Common Measure. Now, for example, the HCF of 60 and 75 is 15, as it is the highest number that can divide 60 and 75 properly. To find HCF, two methods are being employed chiefly, prime factorisation and division. Similar to the LCM, HCF is another vital mathematical application that is often used while calculating fractions to make equations simple and concise.
3. What are the Methods of Solving an Equation?
Ans. There are three methods that one can use to solve any equation. These are substitution, elimination and augmented matrices. Among these three, substitution and elimination are widely used to solve equations, as they allow it to happen in two simple steps. On the other hand, augmented matrices involve a long process, and it has a wide variety of applications compared to the other two. A notable advantage of augmented matrices is that it can solve a system of 3-4 equations. The other two methods are difficult to employ or will make the solution process lengthy.