Step-by-Step Guide: Simplifying Maths Equations
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.
What is a Linear Equation?
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.
Types of Linear Equation
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.
What is Reducing Equation?
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.
Tactics of Simplification
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.
FAQs on How to Reduce Equations to Simpler Form
1. What does it mean to reduce a linear equation to a simpler form?
Reducing a linear equation to a simpler form means performing a series of algebraic operations to make it easier to solve. The primary goal is to isolate the variable on one side of the equation. This process typically involves removing brackets, combining like terms, eliminating fractions or decimals, and arranging the equation into a standard format like ax + b = c.
2. What are the key steps to simplify an equation with variables and constants on both sides?
To simplify an equation with terms on both sides, follow these steps:
- Use the Distributive Property: First, remove any parentheses by multiplying the term outside the bracket with each term inside it.
- Combine Like Terms: On each side of the equation, combine all variable terms together and all constant terms together.
- Transpose Variables: Move all terms containing the variable to one side of the equation (e.g., the left side) using inverse operations.
- Transpose Constants: Move all constant terms to the opposite side.
- Solve for the Variable: Finally, isolate the variable by dividing both sides by its coefficient.
3. Why is finding the LCM essential when reducing an equation with fractions?
Finding the Least Common Multiple (LCM) of the denominators is a crucial step for simplifying equations with fractions. By multiplying every term in the equation by the LCM, you effectively eliminate all denominators. This transforms the complex fractional equation into a simpler one with only integer coefficients, which is significantly easier to work with and reduces the chances of calculation errors.
4. How does simplifying an equation first make it easier to verify the final solution?
Simplifying an equation first streamlines the entire problem-solving process. When an equation is in its simplest form, the final steps to find the variable's value are straightforward. This clarity is beneficial during verification. To verify the solution, you substitute the value back into the original, complex equation. A well-simplified process ensures that the value you found is correct, making the check more of a confirmation rather than another complex calculation.
5. What is the importance of the distributive property in simplifying equations?
The distributive property, which states that a(b + c) = ab + ac, is fundamental to simplifying equations because it is the primary method for removing parentheses or brackets. Complex equations often contain expressions grouped within brackets. Applying the distributive property is typically the first step to break these groups apart, allowing you to then combine like terms and isolate the variable.
6. How can an equation with decimals be reduced to a simpler form without decimals?
To eliminate decimals, you can multiply every term in the equation by a power of 10 (such as 10, 100, or 1000). The specific power of 10 you should use depends on the term with the most decimal places. For example, if an equation has terms like 0.5x and 2.25, you would multiply the entire equation by 100 to convert all decimal terms into integers (50x and 225), making it much simpler to solve.
7. What are some common mistakes students make when reducing equations to a simpler form?
Students often make a few common mistakes. One major error is in applying an operation to only one term instead of every term on both sides of the equation. Other frequent mistakes include:
- Sign Errors: Forgetting to change the sign of a term when moving it to the other side of the equation (transposing).
- Incorrect Distribution: Failing to multiply the outside term by every term inside the parentheses.
- Combining Unlike Terms: Accidentally adding or subtracting terms with variables and constant terms (e.g., adding 3x and 5 to get 8x).
8. Is it always necessary to reduce an equation to its simplest form before solving?
While it is technically possible to solve some equations without full simplification, it is highly inadvisable. Reducing an equation to its simplest form is a strategic step that minimises complexity and prevents errors. Working with large numbers, fractions, or multiple brackets simultaneously increases the risk of mistakes. Simplification establishes a clear, methodical path to the solution, which is a core principle of algebraic problem-solving.
