A linear equation is any equation that contains one or more unknowns where each side of the equation is a sum of coefficients multiplied by those unknowns plus perhaps a constant.

In general, it is good practice to write a linear equation in a normal form where:

1. All scalar multiples of unknowns are collected on the left-hand side of the equation, and
2. all constants are collected on the right-hand side.

Thus, while $3.2x + 5.5y - 6.4 - 1.3z = x - 5.2z - 4.9 + 8.1y + 7.6x$ is a linear equation, in general, you would write such an equation as

$-5.4x - 2.6y + 3.9z = 1.5$.

Describe an algorithm for converting a general linear equation into one that is in normal form.

Solving $ax = b$ when $a \ne 0$

Describe an algorithm that solves $ax = b$ for the unknown variable $x$ where $a$ and $b$ are real numbers and where $a \ne 0$.

As an aside, describe the possible solutions to $ax = b$ when $a = 0$ but $b \ne 0$, and when $a = b = 0$.

A system of two linear equations and two unknowns

Describe an algorithm that determines if the system of linear equations has no solutions (e.g., $x + y = 1, 2x + 2y = 0$), exactly one solution, or infinitely many solutions (e.g., $x + y = 1$, $2x + 2y = 2$). Assume that the unknowns are $x$ and $y$ and that all other values are real numbers:

$a_{1,1}x + a_{1,2} y = b_1$
$a_{2,1}x + a_{2,2} y = b_2$

Next, assuming a unique solution exists, describe an algorithm that finds a solution to the system of linear equations when $a_{1,1} \ne 0$.

Still assuming a unique solution exists, describe an algorithm that finds a solution to the system of linear equations when $a_{1,1} = 0$.

Describe an algorithm for finding all solutions when infinitely many solutions exist.

Finally, combine all these algorithms into a single algorithm that tells you how many solutions exist, and if at least one solution exists, how to find the solutions.

Isolating a variable

Given a linear equation such as

$\frac{3}{5}x+\frac{4}{7}y-\frac{1}{8}z+\frac{8}{3} = \frac{7}{3}x-\frac{2}{3}y+\frac{1}{2}z+\frac{5}{9}$,

we can isolate specific variable. For example, if we isolated this linear equation with respect to each of the three variables, we would get

$x = \frac{5}{7}y-\frac{75}{208}z+\frac{95}{78}$
$y = \frac{7}{5}x+\frac{105}{208}z-\frac{133}{78}$
$z = -\frac{208}{75}x+\frac{208}{105}y+\frac{152}{45}$

Describe an algorithm to isolate a specific variable in a system of linear equations. The equation may not depend on the variable in question, and you should be able to determine when this occurs. For example, $3x + y + 1 = 3x - 3$ does not depend on $x$.