Literal Equations

Literal equations have two or more variables. While solving literal equations, the goal is to isolate one variable in terms of the others. 

If we have the equation of area, A = l × w, and we need to find the value of l (length), we can solve it as: l = A ÷ w

Here, variable l is isolated and expressed in terms of the other variables. 
 

There is no formula to solve all literal equations, because they are all different. A literal equation can be linear, quadratic, or even cubic. To solve the literal equations, we make one letter the subject by writing it in terms of the other letters.

A literal equation typically involves two or more variables. For example, x + x² + 1 = 0 is not a literal equation as it contains only x, whereas ax = b (with variables a, b, x) is a literal equation.
 

Literal equations are used in many formulas, especially in science and math. These formulas are used to find quantities and measurements, such as time, speed, area, or volume. Let us look at some simple examples:

• Mass-Energy Formula

E = mc2
This popular equation has three variables: E (energy), m (mass), and c (speed of light). 

• Area of a Circle

A = 𝜋r2
The formula to calculate the area of a circle has two variables: A (area) and r (radius). 

• Volume of Sphere

V = (4/3)𝜋r3
This formula, which is used to find the volume of a sphere, has two variables: r for radius and V for volume.

• Algebra Example

x + y = 1
This equation has two letters, x and y. It is a simple example of algebra.

In all the above formulas, the letters stand for different things, and we can solve the equation to find the value of one letter using the others. This is why it is called a literal equation. 
 

Literal equations can be solved by isolating one variable and writing it in terms of the others. Sometimes we need to derive the formula for finding a variable. Steps to solve literal equations are:

 

Step 1: Use the inverse operations to move a variable, term, or number to the other side.
 

Step 2: Repeat the process until the variable you want is separated.
 

Step 3: Write the final answer, making sure the chosen variable is the subject of the equation. 

Example: Solve for L in the equation \(P = 2L + 2W\)

Solution: Let us follow the steps below to solve the equation:

Step 1: Move the other terms using the inverse operations.
We need to solve for L, so retain ‘2L’ on the right-hand side, and move the other terms to the left-hand side (LHS).  
\(P - 2W = 2L\)

Step 2: Divide both sides by 2 to separate L

\(\frac{(P - 2W)}{ 2} = L\)

Step 3: The final answer is, \(L = \frac{(P - 2W)}{2}\)
 

Mastering literal equations helps students become comfortable rearranging formulas. Here are some simple and effective tips and tricks to help students confidently handle literal equations.

• Understand the meaning of each variable: Before rearranging, identify what each variable represents in the equation. Knowing what the symbols stand for (like A for area or r for radius) helps avoid confusion and gives the equation real-world meaning.
  
  
   
• Treat variables like numbers: Don’t let multiple letters intimidate you! Solve literal equations the same way you solve regular equations, perform the same operation on both sides. Whether it’s dividing, multiplying, adding, or subtracting, the algebraic rules remain the same.
  
  
   
• Isolate the variable step by step: Focus on the variable you need to solve for and move other terms to the opposite side gradually. Breaking the process into smaller, clear steps reduces errors and improves accuracy.
  
  
   
• Use inverse operations wisely: Remember that inverse operations 'undo' each other. If a variable is multiplied by something, divide to isolate it. If it’s added, subtract it. This logic keeps equations balanced and simplifies complex formulas easily.
  
  
   
• Keep the equation balanced: Whatever operation you do on one side, do the same on the other. Think of it like a balanced scale, if you remove or add weight to one side, you must do the same to the other to maintain equality.

Literal equations are used to rearrange formulas to solve for any variable in terms of the others. These are useful in real-life situations in fields like science, construction, travel, or finance.

• Science: \(E = mc^2\) is a well-known formula. If a scientist knows the value for energy and wants to find the formula for mass, they can rearrange the equation as \(m = \frac{E}{c^2}\).
  
  
   
• Travel and Speed Calculations: If we know the distance and time and want to find the speed, we can rearrange the distance formula \(D = rt\) as \(r = \frac{D}{t}\). If we know the speed and want to find how long the trip will take, we can rewrite it as \(t = \frac{D}{r}\).
  
  
   
• Designing: Architects and engineers rearrange the formulas according to their requirements when designing rooms, buildings, or plots of land.
  
  
   
• Finance: In finance, banks and customers use literal equations to calculate loan interest or savings account growth. 
  
  
   
• Tracking grades or GPA calculations: In schools, the formula for calculating a Grade Point Average(GPA) can be written as: \(GPA =\frac{Total grade points}{total credits}\). This is a literal equation because it involves multiple variables. Students can rearrange the equation to: Total grade points = GPA × Total credits.