Algebra is a branch of mathematics in which letters and symbols are used to represent numbers and quantities. Instead of saying "a number added to 5 gives 9", we can say "x + 5 = 9" and find that x is 4. The use of letters makes mathematical statements shorter, clearer and more powerful, because one statement can describe many different situations at once.
In daily life, there are many quantities whose values are unknown. The number of students in the next class, the price of a book, the weight of a bag, or the height of a tree are all unknown quantities. Algebra gives us a language to talk about these unknown quantities. Letters such as x, y, a, b and n are called variables because their values can change or vary.
In this chapter, we learn about variables, constants, algebraic expressions, how to form expressions from statements, and how to find the value of an expression when the variable is given a number. We also learn about equations and how to solve them. Algebra builds on our knowledge of arithmetic and makes it general, so it is often called generalised arithmetic.
A variable is a quantity that can take different values. Variables are usually represented by letters such as x, y, z, a, b and n. For example, the number of marbles in a bag can be represented by n, which can be any number.
A constant is a quantity that has a fixed value, such as 5, 10, or 3.14. In the expression 5x + 3, the number 5 (multiplier of x) is a coefficient, x is a variable, and 3 is a constant.
Why use variables? Because many rules in mathematics hold for any number. For example, the perimeter of a square = 4 x side. If we call the side s, then the perimeter is always 4s, no matter what s is. This is much shorter than writing the rule in words every time.
Variables help us write general rules for patterns.
A variable allows us to write a single formula that works for many cases.
An algebraic expression is a combination of constants and variables using operations of addition, subtraction, multiplication and division.
Examples of expressions are: 2x + 3, x - 5, 4y, a + b, and x divided by 3.
A term is a single part of an expression separated by + or - signs. In the expression 2x + 3, there are two terms: 2x and 3.
The number multiplied by a variable in a term is called its coefficient. In 5x, the coefficient of x is 5. In -3y, the coefficient of y is -3.
Terms having the same variable raised to the same power are called like terms. Terms with different variables are called unlike terms. For example, 3x and 5x are like terms, but 3x and 3y are unlike terms.
We can add and subtract only like terms. For example, 3x + 5x = 8x, but 3x + 5y cannot be simplified further.
To form an expression, we translate a verbal statement into symbols.
The order is important. "5 subtracted from y" means y - 5, not 5 - y.
An equation is a statement of equality between two expressions. An equation always has an equals sign. For example, x + 3 = 9 and 2y = 12 are equations.
The value of the variable that makes the equation true is called the solution of the equation. For x + 3 = 9, the solution is x = 6 because 6 + 3 = 9.
One method to solve an equation is trial and error: try different values of the variable until the equation becomes true. For 2y = 12, try y = 5: 2 x 5 = 10, not 12. Try y = 6: 2 x 6 = 12, which is true. So y = 6.
We can also solve equations by adding or subtracting the same number on both sides, or multiplying or dividing both sides by the same number. The two sides of an equation must always stay balanced, like the two pans of a weighing balance.
For x + 3 = 9, subtract 3 from both sides: x = 6.
To find the value of an expression, we substitute the given value of the variable and perform the operations.
For example, the value of 5x + 3 when x = 2 is 5 x 2 + 3 = 10 + 3 = 13.
The value of an expression changes when the value of the variable changes. This is why we call the letters variables.
Some expressions involve more than one variable. For example, a + b, 2x + y, and xy - 3. To find the value of such expressions, we substitute the values of all the variables.
For example, if a = 2 and b = 3, then a + b = 2 + 3 = 5, and ab = 2 x 3 = 6.
Algebra is used to express general rules and solve problems: - The total cost of n pens at 10 rupees each is 10n rupees. - The distance travelled in t hours at 50 km per hour is 50t km. - The age of a person 5 years from now, if his present age is a years, is a + 5.
| Statement | Expression |
|---|---|
| x added to 7 | x + 7 |
| 5 subtracted from y | y - 5 |
| 6 multiplied by m | 6m |
| p divided by 4 | p/4 |
| Twice x | 2x |
| 3 more than twice x | 2x + 3 |
| Term | Meaning | Example |
|---|---|---|
| Variable | A quantity that can change | x, y, n |
| Constant | A fixed quantity | 3, 5, 10 |
| Coefficient | Number multiplying a variable | In 5x, 5 is the coefficient |
| Expression | Combination of variables and constants | 2x + 3 |
| Equation | Statement of equality | x + 3 = 9 |
Algebra introduces a powerful symbolic language that lets us generalise arithmetic and solve problems with unknown quantities. We learnt about variables and constants, formed and simplified expressions, translated statements into equations, and solved simple equations. Algebra is the foundation of higher mathematics and is used throughout science, economics and technology. The ability to think with symbols is one of the most important skills we develop in mathematics.