Mathematics can give students a hard time when it comes to solving complex equations. Students often get lost in the several steps that have to be solved before reaching the correct answer and, therefore, need the constant guidance of teachers and parents to be able to achieve the desired results. This is why our web pages have been designed to provide students with assistance on important topics in Mathematics in an efficient and effective manner. Read on to find out everything you need to know about how to solve linear equations with fractions.
However, before we move on to discuss the steps that are required to solve linear equations with fractions, we must first understand the terms individually and separately. Let us understand what each of these terms means. We also need to learn the math of numbers and how it can help us solve linear equations. It is time to befriend numbers instead of shying away from them.
Linear equations are those equations of the first order that represent lines of a coordinate system. Therefore, in other words, linear equations are equations of a straight line and are formulated by y=mx+b, where m represents the slope of the line while b stands for the y-intercept.
However, it might also be wise here to look at the history of linear equations and how they evolved to become such an integral component of Mathematics. The evolution of linear equations is very closely linked to the studies and developments in linear algebra. The earliest studies of linear equations can be traced back to have been done by the European mathematician René Descartes in 1637 after coordinates were introduced in geometry. These changes in geometry led to the rise of a new kind of mathematical geometry that was termed Cartesian geometry. Since lines and planes were important elements in this kind of geometry, there was an urgent need to devise equations to represent the same. This is how linear equations came into being and gradually developed to form a complex branch of mathematics with different systems of such equations existing at their intersections that need to be solved.
Now that we have understood what a linear equation is, it is also important to go through the definition of a fraction to get a better grasp of the topic.
Just like in real life, a fraction is a small portion of a larger piece. In mathematics, a fraction is a value that represents parts of a whole value. These parts hold an equal value that constitutes the whole and is termed numerator and denominator for the top and the bottom value, respectively. The former represents the value of the parts taken, while the former stands for the total number of equal parts in the whole value.
Once again, it is only advisable to be aware of the history of fractions before we proceed further. It is fascinating to learn that work on fractions goes back to the ancient Egyptian civilization. The first evidence of a study on fractions by several Egyptian mathematicians appears around 1600 B.C. in the Rhind Papyrus. However, the fractions that we find in these ancient works are different from our own understanding of fractions. These mathematicians treated fractions more as ratios and unit fractions.
Fractions were also studied and worked on by mathematicians living in ancient India. This version of fractions is closer to how we present fractions today, and is believed by many mathematicians to be the origin from which modern fractions have evolved. The first depiction of fractions in ancient India is recorded to have been done by Brahmagupta in A.D. 630, which was done by writing the numerator and denominator in separate lines without the bar.
The bar in fractions is believed to have originated from the Arabs, who used the bar due to constraints of tying innovations at the time, while the numerator and denominator were differentiated by Latin mathematicians. Until the sixteenth century, multiplication was applied to find the common denominator, before which adding and subtracting fractions had been employed for the same function since the seventh century. Division of fractions was added much later and has continued to date as a common operation of fractions, and might have been the first way to look for a common denominator.
Today, the way to find the common multiple in fractions differs largely from these older operations. But to better understand fractions and its function today, we need to next be well-versed with what a solution is in order to begin solving linear equations with fractions.
Mathematically, a solution is the process of assigning values to variables in an equation that can result in the equation holding true. This means that when a solution is applied to an equation, both sides become equal in value as denoted by the ‘=’ symbol.
Steps to Solve Linear Equations with Fractions
Linear equations can be solved in five simple steps that, when applied to an equation, results in a solution to hold both sides equal. These steps are:
- Clear the fractions in the equation by multiplying both sides with the Least Common Denominator (L.C.D).
- Remove parentheses on each side by using the Distributive Property formula, x(y+z)=xy+yz.
- Combine like terms on each side.
- Undo addition or subtraction present in the equation.
- Undo multiplication or division present in the equation to make the coefficient of the variable equal.
- Undoing these actions simplifies both sides of the equation.
- Solve the equation by isolating the variable on one side and the constant on the other side.
Tips for Solving Linear Equations with Fractions
While we have labelled out the steps that can be applied to solve linear equations in the above section, there are also some tips and points that will be useful for students to remember while attempting to solve linear equations with fractions.
- Any changes made on one side of the equation also have to be made on the other side since the left side is always equal to the right side in an equation.
- Single-variable equations can be solved by isolating the unknown variable on one side to find a number that is equal on the other side.
Things to Remember
Many students can make common mistakes, such as not multiplying both sides by the Least Common Denominator while solving linear equations with fractions. To avoid making such mistakes, here the things you should keep in mind:
- In case of an impossible equality in an equation like x=0, there will be no solution.
- Solutions are always real for equations where the equality holds true in the case of every possible solution.
- To avoid or cancel denominators in an equation, multiply the entire equation with the Least Common Denominator.
- In an equation, parentheses can be removed by multiplying the coefficient present before them to the elements contained within them.
- In the case of a nested parentheses, which refers to a parenthesis inside another parenthesis, the exterior parenthesis is removed first by multiplying whatever value is contained in it with the coefficients.
Frequently Asked Questions
Here are the questions that are frequently asked by students studying linear equations and fractions:
Name the three forms of linear equations.
The three forms of linear equations are standard form, slope-intercept form, and point-slope form.
What is the formula for representing the standard form of a linear equation?
The standard form of linear equations is given by:
Ax + By + C = 0,
where A, B, and C are constants, x and y are variables, and A ≠ 0 as well as B ≠ 0.
How to represent the slope form of linear equations?
Slope form of linear equations is represented by:
where m stands for the steepness of line while c is the y-intercept.
What is the difference between linear and nonlinear equations?
A linear equation represents straight lines, whereas a nonlinear equation does not form a straight line and can be a curve that has a variable slope value.
What are the seven types of fractions?
The seven different types of fractions are – Proper fractions, Improper fractions and Mixed fractions, like fractions, unlike fractions, equivalent fractions, and unit fractions.
How would you define proper fractions?
A fraction where the numerator is smaller than the denominator is called a proper fraction.
How would you define improper fractions?
A fraction is called an improper fraction when the numerator is greater than the denominator.
What is a mixed fraction?
A fraction that contains a combination of a whole number and a fraction is called a mixed fraction.