Literal and General Equations

Definition of Literal Equations

An equation for the variable $x$ in which all the coefficients of $x$, including the constants, are pronumerals is known as a literal equation.

Literal equations are solved by isolating the variable (usually $x$) in terms of other pronumerals. The solution will be an expression involving these pronumerals, rather than a numerical value.

Example:

Solve for $x$ in the literal equation $ax + b = c$.

$$ax + b = c$$
$$ax = c - b$$
$$x = \frac{c - b}{a}$$ (assuming $a \neq 0$)

Solving Linear Literal Equations

The methods for solving linear literal equations are the same as those used for solving standard linear equations with numerical coefficients. The goal is to isolate the variable on one side of the equation.

Steps:

1. Expand brackets: If the equation contains brackets, expand them using the distributive property.
2. Collect terms with $x$ on one side: Rearrange the equation so that all terms containing $x$ are on one side and all other terms are on the other side.
3. Factor out $x$*: If necessary, factor out $x$ from the terms on one side of the equation.
4. Divide to isolate $x$*: Divide both sides of the equation by the coefficient of $x$ to solve for $x$.

Example:

Solve for $x$ in the equation $mx + n = nx - m$.

$$mx + n = nx - m$$
$$mx - nx = -m - n$$
$$x(m - n) = -(m + n)$$
$$x = -\frac{m + n}{m - n}$$ (assuming $m \neq n$)

Solving Simultaneous Linear Literal Equations

Simultaneous literal equations involve two or more equations with two or more variables, where the coefficients are pronumerals. These can be solved using substitution or elimination, just like standard simultaneous equations.

Methods

• Substitution: Solve one equation for one variable and substitute that expression into the other equation(s).
• Elimination: Multiply one or both equations by a factor so that the coefficients of one variable are equal or opposite. Then, add or subtract the equations to eliminate that variable.

Example:

Solve the following simultaneous equations for $x$ and $y$:

$$ax - y = c$$
$$x + by = d$$

1. Multiply the first equation by $b$:

   $$abx - by = bc$$
   2. Add this to the second equation:

   $$abx - by + x + by = bc + d$$
   $$x(ab + 1) = bc + d$$
   $$x = \frac{bc + d}{ab + 1}$$

2. Substitute the value of $x$ into the second original equation:

   $$\frac{bc + d}{ab + 1} + by = d$$
   $$by = d - \frac{bc + d}{ab + 1}$$
   $$by = \frac{d(ab + 1) - (bc + d)}{ab + 1}$$
   $$by = \frac{abd + d - bc - d}{ab + 1}$$
   $$by = \frac{abd - bc}{ab + 1}$$
   $$y = \frac{abd - bc}{b(ab + 1)}$$
   $$y = \frac{ad - c}{ab + 1}$$

Solving Non-Linear Literal Equations

Non-linear literal equations can include quadratic, cubic, or other higher-order terms. Solving these equations may involve factoring, using the quadratic formula, or other algebraic techniques.

Example:

Solve for $x$ in the equation $x^2 + kx + k = 0$.

Using the quadratic formula:

$$x = \frac{-k \pm \sqrt{k^2 - 4(1)(k)}}{2(1)}$$
$$x = \frac{-k \pm \sqrt{k^2 - 4k}}{2}$$

A real solution exists only for $k^2 - 4k \geq 0$, which means $k \geq 4$ or $k \leq 0$.

General Solutions of Equations Involving a Single Parameter

Equations involving a single parameter require you to express the solution in terms of that parameter. This often involves rearranging the equation and considering any restrictions on the parameter.

Example:

Solve for $x$: $a(x + b)^3 = c$

$$(x + b)^3 = \frac{c}{a}$$
$$x + b = \sqrt[3]{\frac{c}{a}}$$
$$x = \sqrt[3]{\frac{c}{a}} - b$$

Example:

Solve for $x$: $x^4 = c$, where $c > 0$

$$x = \pm \sqrt[4]{c}$$

Note: Care must be taken with even powers; for example, $x^2 = 2$ is equivalent to $x = \pm \sqrt{2}$.