Elimination/Examples

Examples of Elimination

Arbitrary Example $1$

Consider the system of simultaneous linear equations:

\(\text {(1)}: \quad\)

\(\ds x + 3 y\)

\(=\)

\(\ds 7\)

\(\text {(2)}: \quad\)

\(\ds 2 x + y\)

\(=\)

\(\ds 9\)

This can be solved by multiplying $(2)$ by $3$ and then subtracting $(1)$ from the resulting equation:

\(\text {(3)}: \quad\)

\(\ds 6 x + 3 y\)

\(=\)

\(\ds 27\)

$(2) \times 3$

\(\ds \leadsto \ \ \)

\(\ds 5 x\)

\(=\)

\(\ds 20\)

$(3) - (1)$

Hence $y$ has been eliminated, and we have:

$x = 4$

from which we can substitute for $x$ in either $(1)$ or $(2)$ and obtain $y = 1$.

Arbitrary Example $2$

Consider the system of simultaneous linear equations:

\(\text {(1)}: \quad\)

\(\ds x + 3 y\)

\(=\)

\(\ds 7\)

\(\text {(2)}: \quad\)

\(\ds 2 x + y\)

\(=\)

\(\ds 9\)

This can be solved by putting $(1)$ in the form:

$x = 7 - 3 y$

from which we can substitute for $x$ in either $(2)$ to obtain:

$2 \paren {7 - 3 y} + y = 9$

from which it follows that $y = 1$ and $x = 4$.