# Systems of Linear Equations

## 10 terms

### system of equations

consists of a number of equations, usually with an equal amount of variables, that are linked in a specific way

### standard form for a system of linear equations

a₁x₁ + a₂x₂ + a₃x₃ + ... = k₁
b₁x₁ + b₂x₂ + b₃x₃ + ... = k₂
c₁x₁ + c₂x₂ + c₃x₃ + ... = k₃
etc.

### elimination

adding or subtracting two equations to eliminate one of the variables

### substitution

two-step process:
1. Solve one of the equations for one of the variable, x or y.
2. Substitute the value of the variable into the other equation.

### When lines are parallel...

...the result is an impossible statement.

### When lines are coexistent...

...the result is a statement that's always true.

### Cramer's Rule

Step 1:
a₁x + b₁y = c₁
a₂x + b₂y = c₂
Step 2:
x = (c₁b₂ - b₁c₂)/d = (c₁b₂ - b₁c₂)/(a₁b₂ - b₁a₂)
y = (a₁c₂ - c₁a₂)/d = (a₁c₂ - c₁a₂)/(a₁b₂ - b₁a₂)

### back-substitution

given a system of three linear equations with three unknown variables, the process of substituting into one of the original equations to solve for the value of the third variable after the system has been solved by reducing the three equations with three variables to a system of two equations with two variables

### ordered triple

the solution of a system consisting of three numbers in parenthesis, separated by commas, where the first value represents x, the second y, and the third z

### decomposing fractions

determining what two fractions came together to create a rational function with polynomials in the numerator and denominator