Systems of equations

Two lines usually cross at one point. That point lies on both, so its $x$ and $y$ make both equations true. Solving a system means finding it.

y = x + 1

and

y = - x + 3

cross at

(1, 2)

Two methods

$y$ is already alone, so replace it in the second equation. $3 x + (2 x + 1) = 11$

Solve. $5 x = 10$ , so $x = 2$

Find $y$ . $y = 2 (2) + 1 = 5$

Answer as a point. $(2, 5)$

Add the equations so $y$ cancels. $2 x = 14$

Solve. $x = 7$

Use it in either equation. $7 + y = 10$ , so $y = 3$

Multiply the second by 2 to match the $x$ terms. $2 x + 2 y = 10$

Subtract. $y = 3$

Then. $x + 3 = 5$ , so $x = 2$

Parallel lines never meet, so no solution. If both equations are the same line, there are infinitely many.

Finding $x$ and stopping.

A solution is a pair. Find $y$ too, then check both equations.

The short version
Use substitution when a variable is already alone. Otherwise multiply so a variable cancels, then add or subtract.

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