About Algebra in Math
What is Algebra?
Algebra is a branch of mathematics that can be considered a unifying thread across almost all of mathematics. In algebra, letters and symbols are used to represent numbers and sometimes even quantities in equations or formulae. Algebra relies on the concept of variables. Variables are basically letters used to represent unknown values.
We can use algebra to figure out the surface area and volume of different types of shapes – 2D and 3D. We can do this by forming equations using algebra. These equations can be used to frame the formulas for volume and surface area.
Branches of Algebra
Algebra, by itself, has multiple sub-branches. They are:
- Elementary algebra
- Advanced algebra
- Abstract algebra
- Linear algebra
- Commutative algebra
Although there are several sub-branches of algebra, we will be looking only at elementary algebra in this article.
Elementary Algebra
Elementary algebra includes numbers and mathematical operations like addition (+), subtraction (-), multiplication (x), and division (÷). However, some of the unknown values might be represented by variables. Typically, we use the variables x, y, z, a, b, and c to represent unknown quantities. If we need more variables, we use other letters too.
When we represent numbers as variables that can carry any value, it helps develop different formulae. For instance, let us take the following formula,
x + y = y + x
This is a law that holds true for a system of all real numbers.
In addition to numbers and mathematical operations, elementary algebra includes the evaluation of expressions and equations, properties of equalities and inequalities, and solutions for linear equations with either one or more variables. Here is an example:
3x = 24
Here, we have the equation 3x = 24. This means that 3 multiplied by some number, represented here as x, is equal to 24.
This is an example of an algebraic equation with one variable, x.
Let us take a look at a couple more examples.
Solved Examples on Algebra
Example 1: What is the solution for 2x = 10 ?
Solution:
Since 2x = 10, we can rewrite this equation as:
x is equal to 10 divided by 2.
x = \(\frac{10}{2}\)
x = 5 [Divide both sides by 2]
Hence, the solution to the equation is x = 5.
Example 2: What is the solution for 2x + 4 = 18 ?
Solution:
Since 2x + 4 = 18, we can rewrite this equation as:
2x = 18 – 4 [Subtract 4 from both sides]
2x = 14
x = \(\frac{14}{2}\) [Divide both sides by 2]
Therefore, x = 7
Example 3: If 5 oranges cost 50 cents, what is the price of one orange?
Solution:
Since 5 oranges cost 50 cents, we can say that 5 multiplied by the cost of one orange is 50 cents. Let us assign the variable x to the cost of one orange. Thus, we get,
5x = 50
This means, x = \(\frac{50}{5}\) [Divide both sides by 5]
Therefore, x = 10
From this, we can conclude that the price of a single orange is 10 cents or a dime.
Example 4: If 10 chairs cost $1,000, what is the price of one chair?
Solution:
Since 10 chairs cost $1000, we can say that 10 multiplied by the cost of one chair is $1000. Let us assign the variable x to the cost of one chair.
Therefore, our equation will now be 10x = 1000
This means, x = \(\frac{1000}{10}\) [Divide both sides by 10]
Thus, we get x = 100
Therefore, we can conclude that the price of a single chair is $100.
Algebra is a common thread that runs across almost all of mathematics. It is what we use in our daily lives to represent and calculate unknown quantities like prices and even to measure the surface areas and volumes of different objects.
A linear equation is a mathematical expression in which all the variables are only to the first power. Therefore, it is also called a first-degree equation.
Yes, it is possible to have an equation with two variables. It will typically take the form ax + by = c, where x and y are variables. Here, the values of a and b cannot be zero. If they are zero, the variables associated with them will also become zero, and then the equation will cease to be a linear equation with two variables.
Yes, we can have a variable in an equation with a power greater than 1. If the power of the variable is 2, the equation is called a quadratic equation. For example, \(2x^2+1=17\). When we solve quadratic equations, we typically get two values for the variable.