Three times the first of three consecutive odd integers is 3 more than twice the third. What is the third integer?
A system was introduced to define the numbers present from negative infinity to positive infinity. The system is known as the Number system. Number system is easily represented on a number line and Integers, whole numbers, natural numbers can be all defined on a number line. The number line contains positive numbers, negative numbers, and zero.
What is an Equation?
An equation is a mathematical statement that connects two algebraic expressions of equal values with ‘=’ sign.
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For example: In equation 3x+2 = 5, 3x+ 2 is the left-hand side expression and 5 is the right-hand side expression connected with the ‘=’ sign.
There are mainly 3 types of equations:
- Linear Equation
- Quadratic Equation
- Polynomial Equation
Here, we will study the Linear equations.
Linear equations in one variable are equations that are written as ax + b = 0, where a and b are two integers and x is a variable, and there is only one solution. 3x+2=5, for example, is a linear equation with only one variable. As a result, there is only one solution to this equation, which is x = 3/11. A linear equation in two variables, on the other hand, has two solutions.
A one-variable linear equation is one with a maximum of one variable of order one. The formula is ax + b = 0, using x as the variable.
There is just one solution to this equation. Here are a few examples:
- 4x = 8
- 5x + 10 = -20
- 1 + 6x = 11
Linear equations in one variable are written in standard form as:
ax + b = 0
- The numbers ‘a’ and ‘b’ are real.
- Neither ‘a’ nor ‘b’ are equal to zero.
Solving Linear Equations in One Variable
The steps for solving an equation with only one variable are as follows:
Step 1: If there are any fractions, use LCM to remove them.
Step 2: Both sides of the equation should be simplified.
Step 3: Remove the variable from the equation.
Step 4: Make sure your response is correct.
Problem Statement: Three times the first of three consecutive odd integers is 3 more than twice the third. What is the third integer?
Let the three consecutive odd integers are num-2, num, num+2, where num is an odd integer.
According to the problem statement, Three times the first of three consecutive odd integers is 3 more than twice the third i.e.
3*(num-2) = 2*(num+2) + 3
To get the numbers, we have to solve this linear equation i.e.
Now, solving the equation using above steps:
3*(num-2) = 2*(num+2) + 3
3*num – 6 = 2*num + 4 + 3
3*num – 6 = 2*num + 7
3*num -2*num = 7 + 6
num = 13
So, the value of num is 13 i.e. the second integer.
First integer is num – 2 i.e. 13 – 2 = 11.
Third integer is num + 2 i.e. 13 + 2 = 15.
So, 11, 13, and 15 are the three consecutive odd integers.
Problem 1: Two times the first number is equal to three times the second number and the sum of both numbers is 5. Find the numbers.
Solution: Let the two numbers are num1 and num2.
According to the problem statement,
Two times the first number is equal to three times of second number i.e.
2*num1 = 3*num2 (eq -1)
Also, Sum of both numbers is 5 i.e.
num1 + num2 = 5 (eq -2)
To get the numbers, we have to solve these equations i.e.
Now, solving the equation using the above steps:
Taking eq-2 :
2*num1 = 3*num2
num1 = (3*num2) / 2
Taking eq-1 i.e.
num1 + num2 = 5
Now put the result of 1st equation i.e. num1 = (3*num2)/2 in 2nd equation i.e.
(3*num2)/2 + num2 = 5
(3*num2 + 2*num2 ) / 2 = 5
3*num2 +2*num2 = 5 * 2
5*num2 = 10
i.e. num2 = 10/5 i.e. 2
So, the value of num2 is 2 and using this the value of num1 is 5-num2 = 5-2 = 3.
Problem 2: The sum of four consecutive numbers is 18, find the numbers.
Let the four consecutive numbers are x, x+1, x+2, x+3 respectively.
So, according to the problem statement:
x + x+1 + x+2 + x+3 = 18
Using this equation we can get the value of x i.e. the first number
4x + 6 = 18
4x = 18-6
4x = 12
x = 12/4
x = 3
So, the numbers should be 3, 4, 5, and 6.