Algebra follows all the rules of Arithmetic. It uses the same four operations on which arithmetic is based, i.e. addition, subtraction, multiplication, and division.
But Algebra introduces one new element. The element of the "unknown".
A constant does not change over time and has a fixed value. For example, 2, 6, 1212, pi. Variables are values that can change over time. For example, the temperature at different times of the day represents a variable. The weight of a student in your grade is a variable, as it varies from student to student.
Example: In 2x, 2 is a constant and x is a variable. In 4 + xy, 4 is a constant, and x and y are variables.
An algebraic expression is a combination of the constants and the variables connected by some or all of the four fundamental operations (+, −, ×, ÷). For example, 2x + 10y + 3 is an algebraic expression. Let us try creating an algebraic expression for the following statement:
"You solved x math questions yesterday. Today you did 10 questions less. How many questions have you solved today?"
The algebraic expression that explains the number of questions solved by you today is
 |
If 4 is a constant and z is a variable then -
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In arithmetic we write 2 + 3 =
In algebra same will be written as 2 + 3 = x
Here
The above expression '2 + 3 = x' is called 'Algebraic Equation'.
An equal sign denotes that the value of the left-hand side is equal to the right-hand side or we can say it is a balanced equation.
In the Algebraic equation, we find the value of a variable. A Variable is a symbol for a number we don't know yet. 'x' is a variable in equation 2 + 3 = x.
Let's understand the variables by taking a few examples.
Students buy notebooks from a bookstore. A notebook cost $5. If n is the number of notebooks that the student wants to buy, then n can take the value like 1, 2, 3, and so on. And the student has to pay \(5n\) price for n number of books. The total cost of n notebooks is given by the rule:
Let us take one more example. Mary has 10 more apples than Jerry. So if Jerry has 'm' number of apples, Mary has '10+m' apples.
In both cases, m is a variable. However, the algebraic expression for both is different.
Let us also see how common rules in mathematics that we have already learned are expressed using variables.
Commutativity of addition of two numbers
We know that 3 + 4 = 4 + 3, therefore x + y = y + x
This is expressing the rule in generic form using variables x and y.
Commutativity of multiplication of two numbers
3 × 4 = 4 × 3, 33 × 23 = 23 × 33 (order in multiplication does not change the result), therefore we can write this rule in variables as x × y = y × x or xy = yx
Distributivity of numbers
7 × 42 can also be written as \( 7\times(40 + 2) = 7 \times 40 + 7 \times 2 = 280 + 14 = 294\),
\(x \times (y + z) = xy + xz\)
Associativity of numbers
As (\(6+3) + 4 = 6 + (3 + 4)\) therefore, we can generically write \((x + y) + z = x + (y + z )\)
Similarly, \((x \times y) \times z = x \times (y \times z)\)
To solve the Algebraic Equation, move unknown values to one side and known values to the other side. Let's take an example and try to find the value of x.
Example 1:
\(x -2 = 3\)
Add 2 to both sides. Please note that addition, subtraction, multiplication, and division by the same number on both sides of the equation don't affect the balancing of the equation, and '=' still holds true. i.e. Left Hand Side = Right Hand Side
\(x - 2 + 2 = 3 + 2\)
⇒ \(x = 5\)
Example 2:
Solve below algebraic equation for x: \(x + 2 = 6\)
Subtracting 2 from both sides. This way we are following the rule to have one side only with unknown and the other side with known values.
\(x + 2 - 2 = 6 - 2\)
⇒ \(x = 4\)
Example 3:
\(4 \times x = 20\)
Divide both sides by 4
\(\frac{4 \times x}{4} = \frac{20}{4}\)
⇒ \(x = 5\)
Example 4:
\( \frac{x}{3}\)= 5
Multiply both sides by 3
\(\frac{x}{3} \times 3 = 5 \times 3\)
⇒ \(x = 15\)
Please note an algebraic equation can have more than one variable.
