Namrata Das Exams Prep Master
Exams Prep Master
Solving equations having variables on both sides is very useful in mathematical expressions. It helps us deal with the unknown quantities present. Variables are symbols denoting quantities which may change. They may stand for a function, or a vector or even a number. When we have variables on both sides of the equation, we get a better depth of the problem which further improves our concepts. Let us take a look at such equations and how to deal with them.
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Also read: Isosceles Triangle Theorems
Definition of Solving Equations Having The Variable On Both Sides
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Solving equations having variables on both sides increases the flexibility of a student’s mind. It helps them by quickly identifying the solution. Students also get to solve the equations by multiplication table and inverse operations. This technique is very effective and helps save a lot of time at hands by exercising the brain to the fullest.
This methodology of solving equations is very clear and precise to understand. It makes it easier for the students to distinguish the constant and variable immediately. It is a detailed process without missing any single step.
Importance of solving equations having the Variable on both sides
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Of everything, there are certain equations in mathematics which we need to solve only by devising variables on both right and left hand sides. Now, what is a variable? In simple terms, a variable is basically a symbol whose function is to represent a quantity in a mathematical form. This variable may change as the problem or experiment increases its pace. Letters x, y, and z are generally used as variables.
This way of solving the equations has many benefits to it like, it increases the speed and efficiency of a student in solving these problems. The more you solve these, the quicker you get better at them. It helps clarify the basic concept of an equation when they solve such problems with the help of detailed worksheets.
Students can do much better as these equations are based or provided on real-life examples. It becomes simpler for the student to grasp it without having any room for doubts.
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Step and Examples
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- The effective way to solve equations with variables on both sides is to first simplify it.
- Get the variable on one side.
- In order to do the previous step, the addition or subtraction sign of the equation must be reversed. It means that we must take a quantity containing the desired variable and either add it to both sides or subtract it from both sides.
- The smaller quantity should always be subtracted from the bigger one as it makes the process easier.
- When the variable is at one side, inverse operations can be carried out to get the final result.
Solved Examples
Ques: 5x + 7x = 14 - 2x. Solve for x.
Solution: Given that, 5x +7x = 14 - 2x
On further simplifying the equation we get,
12x = 14 - 2x
12x + 2x = 14
14x = 14
x = 14/14
x = 1
We can check this by substituting the value we got for x in the original equation,
5(1) + 7(1) = 14 - 2(1)
12 = 12? Yes, they are the same.
Ques: Find the value of x in the given equation. 5x + 10 = -20.
Solution: Given that, 5x + 10 = -20
We will reverse the equation by changing signs,
5x + 10 = -20 - 10
5x = -30
x = -30/5
x = -6
We can check this by substituting value in original equation,
5(-6) + 10 = -20
-30 + 10 = -20
-20 = -20?
Yes, they are the same.
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Things to Remember
- Solving equations having the variable on both sides makes it easier for students to understand the basic concepts. The student can understand them better with the provided detailed step by step working.
- This method of solving equations is very simple and exercises the brain. By practicing such problems students can easily identify the technique and solve it by using multiplication tables and inverse operations.
- Variable is a symbol which represents a quantity in mathematics. We usually take x, y, z as popular variables in algebra. They have made problems fun to solve by clearing the basic fundamentals of students. This variable may change further.
- Proper steps must be followed while solving equations having the variable on both sides. One must simplify it and then get the variable to one side. This makes it quicker to solve.
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Sample Questions
Ques. Define variables. How is it different from a constant? (2 marks)
Ans. A variable is a symbol, or a single letter which denotes a quantity in mathematics. This symbol may change further. Letter x, y, z of the alphabet are commonly used for representing such variables. A constant, on the other hand, is totally opposite in nature. It is a letter or a numeric value which is fixed. It is a known value whereas the variable is unknown.
Ques. Solve for x in the given equation. -2x = -8 - 3x (3 marks)
Ans. Given that, -2x = -8 - 3x
Now, upon simplifying the equation we get,
3x - 2x = -8
x = -8
Moving on,
x + 8 = -8 + 8
On the right hand side both get cancelled and we are left with,
x + 8 = 0
x = -8
Ques. Why do we need variables? (2 marks)
Ans. variables are an important aspect of algebra in mathematics. Without them, it would be very difficult to solve even the simplest of equations. They indicate certain unknown values and are used to store information regarding the equation. For example, x + 2 = 8. It means x is 6.
Ques. Solve for y in 6y - 24 = 12y. (4 marks)
Ans. Given that 6y - 24 = 12y
Upon simplifying this equation we get,
6y = -24
Now we will try to bring the variable on one side,
6y + 24 = -24 + 24
6y = -24
y = -4
We can also check this resultant value by putting it in the original equation.
6(-4) - 24 = 12(-4)
-48 = -48
As LHS=RHS, y = -4
Ques. What are the types of variables? (3 marks)
Ans. The variables in mathematics are divided into two broad categories. These are known as the independent and the dependent variable. The quality of a dependent variable depends on the estimation of another variable. Whereas the values of independent variables are independent of changes. If the equation gives y = x2, it means x is independent whereas y is dependent.
Ques. Give some benefits of solving equations having variables on both sides. (3 marks)
Ans. There are certain perks of solving an equation having variables on both sides. These are,
- Easy and precise to understand
- Clear up the basic fundamentals
- Detailed explanation with step by step analysis
- Saves time with inverse operations
Ques. What are inverse operations? (3 marks)
Ans. Inverse operations are operations which reverse the effect of the other operation. These operations may be addition, subtraction, multiplication or division. These are extremely helpful during solving equations. For example, if we are adding two numbers like 7 + 4 = 11, then the inverse of these addition operations will be subtraction operations. This can be written as 7 - 4 = 3.
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