Question

How do you solve \[x - \dfrac{1}{2} = 1\dfrac{1}{4}\] ?

Answer 1

Hint: Here, the given question is in the form of an algebraic linear equation or expression. The unknown variable is determined by using a multi-step equation. The constant terms in the given equation are in the form of proper fraction and mixed fraction. Mixed fraction needs to convert improper fraction and then solve for x.

Complete step-by-step answer:
A linear equation in one variable is an equation that can be written in the form \[ax + b = c\] , where a, b, and c are real numbers and \[a \ne 0\] . Linear equations are also first-degree equations because the exponent on the variable is understood to be 1.
To solve x in a given system of linear equations by using multi-step equations. Multi-step equations are algebraic expressions that require more than one operation, such as subtraction, addition, multiplication, division, or exponentiation, to solve.
Consider the given equation.
  \[ \Rightarrow x - \dfrac{1}{2} = 1\dfrac{1}{4}\]
Convert mixed fraction \[1\dfrac{1}{4}\] to an improper fraction by multiplying the denominator by the whole number 1, and adding the numerator 1. Set the result over the denominator 4. Then
 \[ \Rightarrow x - \dfrac{1}{2} = \dfrac{5}{4}\]
Add both side by \[\dfrac{1}{2}\] , then
 \[ \Rightarrow x - \dfrac{1}{2} + \dfrac{1}{2} = \dfrac{5}{4} + \dfrac{1}{2}\]
 \[ \Rightarrow x = \dfrac{5}{4} + \dfrac{1}{2}\]
The two fractions in above equations are unlike fractions.
When we add and subtract two unlike fractions, we have to make the denominator equal first and then perform the respective operation. There are two methods by which we can make the denominator equal. They are: Cross-Multiplication Method and LCM Method
In the cross-multiplication method, we cross multiply the numerator of the first fraction by the denominator of the second fraction. Then multiply the numerator of the second fraction by the denominator of the first fraction. Now, multiply both the denominators and take it as a common denominator. Then we can add or subtract the fractions now. i.e.,
 \[\dfrac{A}{B} + \dfrac{C}{D} = \dfrac{{AD + BC}}{{BD}}\] and \[\dfrac{A}{B} - \dfrac{C}{D} = \dfrac{{AD - BC}}{{BD}}\]
Then
 \[ \Rightarrow x = \dfrac{{10 + 4}}{8}\]
 \[ \Rightarrow x = \dfrac{{14}}{8}\]
To write the simplest form of fraction using a HCF of both numerator and denominator.
Factors of 14 are 1, 2, 7 and 14
Factors of 18 are 1, 2, 4 and 8
The highest common factors HCF between 14 and 18 is 2
Divide both numerator and denominator by 2.
 \[\therefore x = \dfrac{7}{4}\]
So, the correct answer is “$ x = \dfrac{7}{4}$”.

Note: The fractions can be solved by using the concept of LCM where LCM is abbreviated as least common factor. Where the number is completely divisible by the denominator of the fraction. This fraction is also solved by using another concept of cross multiplication. It is just cross multiplying.