Non-Linear Equations: Background You’ll Need 2

  • Rearrange formulas to solve for one variable

Many real-world applications (problems that can be modeled by a mathematical equation and solved for an unknown quantity) involve formulas that describe relationships between quantities.

Examples of formulas that commonly occur in applications include:

  • the perimeter of a rectangle of length L and width W
    • [latex]P=2L+2W[/latex]
  • the area of a rectangular region of length L and width W
    • [latex]A=LW[/latex]
  • the volume of a rectangular solid with length L, width W, and height H
    • [latex]V=LWH[/latex]
  • the distance [latex]d[/latex] covered when traveling at a constant rate [latex]r[/latex] for some time [latex]t[/latex]
    • [latex]d=rt[/latex].

Formulas such as these may be used to solve problems by substituting known values and solving for an unknown value. You should know these formulas and be able to recognize when to apply them to a problem.

We often need to rearrange formulas to isolate a particular variable. This process is crucial for solving equations and expressing relationships between variables in a more useful form.

How to Isolate a Variable:

  1. Identify the target variable: Determine which variable you want to isolate.
  2. Use inverse operations: Apply the opposite of each operation affecting your target variable, working from the outside in.
  3. Perform the same operations on both sides: Remember, what you do to one side of the equation, you must do to the other to maintain equality.
  4. Simplify: Combine like terms and simplify expressions as you go.
Isolate the variable for width [latex]w[/latex] from the formula for the perimeter of a rectangle:  

[latex]{P}=2\left({l}\right)+2\left({w}\right)[/latex]

Isolate the variable for height, [latex]h[/latex], from the formula for the surface area of a cylinder,

[latex]s=2\pi rh+2\pi r^{2}[/latex]