Simultaneous Equations

Make the coefficients of one variable the same, then add or subtract.

$2x + 3y = 12$ ... (1) $4x - 3y = 6$ ... (2)

Add: $6x = 18$ → $x = 3$. Substitute: $6 + 3y = 12$ → $y = 2$.

Rearrange one equation and substitute into the other.

$y = 2x + 1$ ... (1) $3x + y = 11$ ... (2)

Substitute (1) into (2): $3x + 2x + 1 = 11$ → $5x = 10$ → $x = 2$, $y = 5$.

$y = x + 3$ ... (1) $x^2 + y^2 = 25$ ... (2)

Substitute: $x^2 + (x+3)^2 = 25$ → $2x^2 + 6x + 9 = 25$ → $2x^2 + 6x - 16 = 0$ → $x^2 + 3x - 8 = 0$

Use the quadratic formula.

The solution is the point where the two lines (or line and curve) intersect.