Simultaneous Equations/Examples/Arbitrary Example 1
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Example of Simultaneous Equations
| \(\text {(1)}: \quad\) | \(\ds x + y\) | \(=\) | \(\ds 6\) | |||||||||||
| \(\text {(2)}: \quad\) | \(\ds 2 x + y\) | \(=\) | \(\ds 4\) |
has the solution:
| \(\ds x\) | \(=\) | \(\ds -2\) | ||||||||||||
| \(\ds y\) | \(=\) | \(\ds 8\) |
This can be interpreted as that the point $\tuple {-2, 8}$ on the Cartesian plane is where the two straight lines $x + y = 6$ and $2 x + y = 4$ intersect.
Proof
| \(\text {(1)}: \quad\) | \(\ds x + y\) | \(=\) | \(\ds 6\) | |||||||||||
| \(\text {(2)}: \quad\) | \(\ds 2 x + y\) | \(=\) | \(\ds 4\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds -2\) | $(2) - (1)$ | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds -2 + y\) | \(=\) | \(\ds 6\) | substituting in $(1)$ | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds y\) | \(=\) | \(\ds 8\) | adding $2$ to both sides |
$\blacksquare$
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): simultaneous equations
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): simultaneous equations