Algebra Problems

Algebra Problems with Solutions

Algebra problems are not just based on the algebraic expressions but also depends on various types of equations in maths in which a quantity or a variable is unknown to us. Many of you would be familiar with the word problems, but are you aware of the fact and the problems that are related to the variables and constants? When you say 5 it means the number 5 but what if you say x = 5 or 5y? This is when algebra came into existence. Algebra is the branch of the Mathematics that not only deals with the numbers but also deals with the variables and alphabets. The versatility of algebra is much deep and conceptual. All the non-numeric characters in algebra represent the variable and numerics represent the constants. We will study in this article about the algebra problems with solutions and take a look at the algebra word problems as well.

Let us first revise on the basic algebra identities.

Algebra Identities

Given below are the algebra identities that you must learn to solve Algebra problems.

1. (a + b)2 = a2 + b2 + 2ab

2. (a - b)2 = a2 - b2 - 2ab

3. a2 - b2 = (a + b) (a - b)

4. a2 + b2 = (a + b)2 - 2ab = (a - b)2 + 2ab

5. a3 + b3 = (a + b) (a2 - ab + b2)

6. a3 - b3 = (a - b) (a2 + ab + b2)

7. (a + b)3 = a3 + 3ab (a + b) + b3

8. (a - b)3 = a3 - 3ab (a - b) - b3

6th Grade Algebra Problems

In class 6, you will be introduced to the algebra concept. Here, you will learn about how the unknown values can be represented in terms of the variables.  The given expression can be solved only when you know the value of the unknown variable.

Consider the following example:

Six less than a number is equal to two. Determine the number

Solution:

Let the given number be denoted as ‘x’.

According to condition, you have x – 6 = 2

By solving it algebraically for x, you get,

x = 2 + 6 = 8

Hence, your required number is 8.

Algebra Mathematics Problems and Solutions

There are different types of terminology that are used in the algebraic equations such as term, factor, and coefficient. Let us understand these terms with an example. 

Consider 4x + 5y to be an algebraic expression, then 4x and 5y are called the terms. Since the variables used here are x and y, hence, x and y are called the factors of 4x + 5y. And the numerical factor that is attached to the variables is the coefficient. 4 and 5 here are the coefficients of x and y respectively in the given expression.

Any expression having one or more terms is known as a polynomial. Specifically, a one-term expression is known as a monomial, a two-term expression is known as a binomial, and a three-term expression is known as a trinomial.

The terms that have the same algebraic factors are the like terms. The terms that have different algebraic factors are called, unlike terms. Hence, the terms 4xy and – 3xy are called like terms, but the terms 4xy and – 3x are called, unlike terms.

Algebra Word Problems

Let us now learn about how to solve the algebra word problems by taking some examples.

1. A man starts his car from Delhi to Amritsar at 6.00 am. Assume the uniform speed of his car to be x km/h. At 12:00 noon, he learns that he is 50 km away from Amritsar. Determine the distance between Delhi and Amritsar.

Solution:

The time taken by the man to reach Amristar = 12:00 noon – 6.00 am = 6 hours

The uniform speed of the car = x km/h

Therefore, the total distance covered by the man = Time x speed = 6x km

Hence, the distance between Delhi and Amristar is (6x + 50) km.

2. The present age of Deepak is one-third of his mother’s present age. If his mother’s age was five times than his age 6 years ago, determine their present ages.

Solution:

Let the present age of Deepak’s mother be x years

Hence, the present age of Deepak = x/3 years

6 years ago, the mother’s age = (x – 6) years

Hence, Deepak’s age 6 years ago = ( \[\frac{x}{3}\] - 6 )years

According to the problem, 6 years ago, the mother’s age was 5 times than that of Deepak’s age.

Hence,(x - 6) = 5 x (\[\frac{x}{3}\] - 6)

Solving this equation, you get

x - \[\frac{5x}{3}\]  = -30 + 6

= \[\frac{3x - 5x}{3}\] = -24

= \[\frac{-2x}{3}\] = -24

= 2x = 24 x 3

Hence, x = 72/2 = 36

Therefore, the present age of Deepak’s mother is 36 years.

The present age of Deepak = x/3 = 36/3 = 12 years.