Literal Equations
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Introduction

    This lesson page will inform you how to solve literal equations, which is a STEM (Science, Technology, Engineering, and Mathematics) topic. Here are the sections within this lesson page:

    Before continuing with this lesson, it is important that you understand how to linear solve equations. Use this before proceeding if you are uncertain how to solve linear equations.

    esson: Solving Linear Equations


    If you want to know what a literal equation is, consider this well known equation.

circumference formula

    This equation is the relationship between the circumference of a circle with its diameter. The equation indicates how to determine the circumference of a circle -- by multiplying its diameter and the value pi. However, what if we knew the circumference of a circle and wanted to know its diameter?

    It would be helpful to know how to calculate the diameter by having a better equation. If we were able to solve the equation for the d-value, the equation would be more handy.

    The process of rearranging terms and performing mathematical operations on an equation to solve for a different variable is called solving a literal equation.

    Likewise, the area of a square is written like so.

area of a square formula

    This means the area of a square is equal to the length of its side times itself. What is the length of a side of a square if you know the area of that square? The answer to that question demands that we are able to solve a literal equation to get the s-value alone.

    Being able to solve literal equations like the ones above is required to fully understand and utilize all mathematical relations, like those found within several science disciplines.

    The next section will continue down that road and demonstrate how to solve literal equations.


    Let us examine a few equations to get a feel for how to solve literal equations.

Example 1: Solve for the diameter.

area of a square formula

    To get the letter-d alone, we need to find a way to remove the pi from the right side. Since pi is being multiplied to the letter-d, the inverse operation would be to divide by pi. So, dividing both sides by pi gives us this.

solving literal equation circumference

    The diameter of a circle is equal to its circumference divided by pi.

Example 2: Solve for the side.

area of a square formula

    The equation is s-squared is equal to area. The inverse of squaring is square-rooting. So, we need to take the square root of both sides to simplify the right side of the equation, like so.

area of a square formula

    This tells us that the square root of a square's area is equal to the length of its side.

    Try this quizmaster to determine if you understand basic literal equations.

    uiz: Solving the Gas Law Relation


    Here are more difficult problems to consider.

Example 1: Solve for the length.

perimeter of a rectangle formula

    This formula indicates exactly what to do with the length and width of a rectangle to gain its perimeter.

    On the other hand, if we want to get the length -- the letter-l alone -- we have to first get the 2w term to the opposite side of the equation. Since we are adding 2w, the inverse is to subtract 2w from both sides, like so.

literal equation perimeter rectangle

    Now, to cancel the two that is being multiplied to the l-variable, we need to divide both sides of the equation by two.

literal equation perimeter rectangle

    This equation now tells us how to use the perimeter of a rectangle given its perimeter and width.

Example 2: Solve for the height.

surface area cylinder formula

    This equation tells us how to find the surface area of a cylinder using its radius and height. Let's use the equation to solve for h, the height of a cylinder. First, we need to do the distributive property.

literal equation surface area cylinder

    Now, we need to get the h-term alone, which means canceling the term with the r-squared. So, we need to subtract that term from both sides.

literal equation surface area cylinder

    For our last step, we need to divide both sides of the equation by 2(pi)r to get the h-variable alone, like so.

literal equation surface area cylinder

    This tells us how to determine the height of a cylinder given its surface area (S) and radius (r).

    The following quizmaster contains six moderate level problems. Give it a look to see if you understand how to manipulate literal equations.

    uiz: Solving Literal Equations


    Try these lessons, which are related to the sections above.

    uiz: Solving the Gas Law Relation
    uiz: Solving Literal Equations


    Try these lessons, which are related to the sections above.

    esson: Graphing Lines
    esson: Systems of Equations
    esson: Solving Inequalities
    esson: Linear Programming
    esson: Conic Sections: General Form to Standard Form