Literal Equations
An extension of solving linear equations is solving literal equations. A literal equation is an equation where variables represent known values. Often times, literal equation are formulas that we would use to represent equations of lines, slope, area, volume, distance, time, etc.
Here are some examples of literal equations:
When we are solving a literal equation, we are solving for one variable in terms another variable or variables. The process you go through to solve a literal equation is the same as solving a linear equation.
Example 1: Solve the literal equation \(y=mx+b\) for \(x\).
An extension of solving linear equations is solving literal equations. A literal equation is an equation where variables represent known values. Often times, literal equation are formulas that we would use to represent equations of lines, slope, area, volume, distance, time, etc.
Here are some examples of literal equations:
- Area of a Trapezoid: \(A=\frac{1}{2}h\left(b_1+b_2\right)\)
- Area of a Circle: \(A=\pi r^2\)
- Slope-Intercept Equation of a Line: \(y=mx+b\)
When we are solving a literal equation, we are solving for one variable in terms another variable or variables. The process you go through to solve a literal equation is the same as solving a linear equation.
Example 1: Solve the literal equation \(y=mx+b\) for \(x\).
\(\begin{align}y&=mx+b \ & \ &\text{1) Rewrite the formula/equation.}\\
y-b&=mx \ & \ &\text{2) Subtract b from both sides (undo the addition with the variable you are solving for).}\\
\frac{y-b}{m}&=x \ & \ &\text{3) Divide both sides of the equation by m (undo the multiplication with the variable you are solving for).}\end{align}\)
y-b&=mx \ & \ &\text{2) Subtract b from both sides (undo the addition with the variable you are solving for).}\\
\frac{y-b}{m}&=x \ & \ &\text{3) Divide both sides of the equation by m (undo the multiplication with the variable you are solving for).}\end{align}\)
Example 2: Solve the literal equation \(A=\frac{1}{2}h\left(b_1+b_2\right)\) for \(b_1\).
Example 3: Solve the literal equation \(A=\pi r^2\) for \(r\).
Example 4: Solve the literal equation \(2y+x=3xy-1\) for \(y\) in terms of \(x\).
\(\begin{align}2y+x&=3xy-1 \ & \ &\text{1) Rewrite the original literal equation}\\2y-3xy&=-x-1 \ & \ &\text{2) Get all y terms on one side and non-y terms on the opposite side}\\y\left(2-3x\right)&=-x-1 \ & \ &\text{3) Factor out the y}\\y&=\frac{\left(-x-1\right)}{\left(2-3x\right)} \ & \ &\text{4) Divide both sides by the expression to isolate y}\end{align}\)
Note: It does not matter to which side you move the y terms--your answer will be the same, but all terms will be opposite in sign.
Example 5: Solve the literal equation \(3x+2xy=y-5\) for \(y\) in terms of \(x\).