Elimination and Cramer's rule | Algebra 2

Adding equations

If two equations are both true, adding them gives another true equation: add the left sides, add the right sides. When one variable has opposite coefficients, adding makes it disappear.

The elimination method

Usually the coefficients aren't opposites yet. Multiply one or both equations by well-chosen numbers to make them opposites.

Determinants and Cramer's rule

There's also a formula that produces the solution directly. It uses a number called a determinant.

For the system $\begin{cases} a_1 x + b_1 y = c_1 \\ a_2 x + b_2 y = c_2 \end{cases}$ compute three determinants:

D = \begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix} \qquad D_x = \begin{vmatrix} c_1 & b_1 \\ c_2 & b_2 \end{vmatrix} \qquad D_y = \begin{vmatrix} a_1 & c_1 \\ a_2 & c_2 \end{vmatrix}

$$D$$ uses the coefficients. For $$D_x$$ , replace the $$x$$ column with the constants; for $$D_y$$ , replace the $$y$$ column.

No solution or infinitely many

When $$D = 0$$ , Cramer's rule can't be used, and the system doesn't have a unique solution. Elimination tells you which case you're in: both variables disappear.

What elimination leaves	Type of system	Solutions	Lines
a single value for each variable	consistent, independent	exactly one	intersecting
a false statement like $$0 = 5$$	inconsistent	none	parallel
a true statement like $$0 = 0$$	consistent, dependent	infinitely many	the same line

Practice

Quick check

What is the determinant $$D$$ of $\begin{cases} 3x - y = 2 \\ x + 2y = 3 \end{cases}$ ?

Quick check

Elimination on a system leaves you with $$0 = 0$$ . The system has: