Solving Equations With Variables On Both Sides

Equations are one of the basics of mathematics, and solving equations with variables on both sides is another important concept in which both the unknown quantity may appear in the equation, and we find the answers of both. So, let us start by describing how to solve equations with variables on both sides.

What is an equation? 

The equation can be defined as an expression that equals two different values in mathematics. The equations are further separated by an equal sign placed in the middle. If the equation has the equal sign (=), it means that the value of the left-hand side should be the same as the value on the right-hand side of the equation. There is either one variable or more than one variable in a linear equation. Here is an example of it – 

x = 20 +15, find the value of “x”. 

Looking at the equation, it can be said that the value generated of the “x” needs to be equal to the number 35 as 20+15 = 35. The process through which the equation will be solved is termed “Solving equations.” 

The Standard Form of an Equation 

Here is the standard form which is used for representing an equation – 

Ax + By = C 

According to this equation, we need to find both x and y. 

Solving Equations With Variables On Both Sides 

At times, both the unknown quantity may appear in the equation. For example, Ax + By = C. Here, the values of x and y both are unknown in an equation. A quantity with variables can be treated the same as that without any variables. For example, the quantity with a variable can be added to both sides without making changes in the equation or values, which makes it true. 

15 – x = 4x

15 – x + x = 4x + x

15 + 0x = 5x

15 = 5x

3 = x

x = 3

A term along with the variable can be subtracted from both the sides of the given equation – 

5x = 6 + 2x

5x – 2x = 6 + 2x – 2x

3x = 6 + 0x

3x = 6

x = 2

Once the equation is simplified, the first step involves solving the equation with the variable on both sides to get the variable on a single side. This can be done by reversing the addition and subtraction signs of one of the terms with the variable. It is mostly easier to add or subtract the smaller quantities from larger quantities to work with the positive coefficients. Once all the variables are on one side, we can consider inverse operations. 

Let us understand solving equations with variables on both sides via an example – 

Example 1 : We have an equation 5x = 12 – x

Get all the variable on the one side:

We now have, 

5x + x = 12 – x + x

6x = 12

Solving the equation using the inverse operation= x = 2

Check: 3(2) + 2(2) = 12 – 2

Example 2 : y: 5y – 3 = 3y + 5

Here, the equation is already simplified.

Get all the variables on one side:

Now we have, 

5y – 3 – 3y = 3y + 5 – 3y

5y – 3y – 3 = 3y – 3y + 5

2y – 3 = 5

Solving the equation using the inverse operation:

2y – 3 + 3 = 5 + 3

2y = 8

=y = 4

Check: 5(4) – 3 = 3(4) + 5 

Linear Equation 

A linear equation can be described as an equation with the highest degree of power of one variable. In simple terms, no variable has an exponent of more than one in a linear equation. In a graph, a linear equation constantly forms a straight line. Hence, it is named the linear equation. 

• A linear equation in which only one variable is present, for example, Mx+N=0, where M and N are constant, and x is a variable.
• A linear equation in which two variables are present, for example, Mx+Ny=O, where M, N, and O are constant and x and y are variables.

Standard form of a Linear Equation

There are three ways in which the standard form of linear equations can be written. These include one variable, two variables, and three variables, and it is also called a general form of linear equation.

1. In one variable: px+q=0, where p and q ≠ 0 and integer. x is a variable.
2. In two variables: px+qy=r, where p, q, and r ≠ 0 are integers. x and y are variables.
3. In three variables: px+qy+rz=s, where p, q, r, and s ≠ 0 are integers. x, y, and z are variables.

Golden Rules of Linear Equation 

1. The outcome of a linear equation cannot be changed if the same number is subtracted, added, divided, or multiplied into both sides of the equation.
2. The value of the variable generating a linear equation valid is named the solution or root of the linear equation. 
3. The graph of a linear equation in one or two variables constantly forms a straight line.

Conclusion 

With this, we come to an end to solving equations with variables on both sides. Sometimes, both the unknown quantity may appear in the equation. A quantity that has variables can be treated the same as the quantity without any variables. 

In this article describing solving equations with variables on both sides, we studied the concept of equations in length. We covered several other topics, such as solving equations with variables on both sides, examples of solving equations with variables on both sides, and other related topics. We hope this study material helped you better understand solving equations with variables on both sides.