An equation like $y = 2x - 1$ has many solutions: every point on its graph makes it true (see slope-intercept form). A system asks for two equations to be true at the same time.
Por qué funciona
The graph of an equation is exactly the set of points that make it true. A point that makes both equations true must therefore lie on both graphs. Two lines with different slopes share exactly one point, where they cross, so that point is the solution.
Idea clave
A solution of a system of two equations is an ordered pair $(x, y)$ that makes both equations true. On a graph, it is the point where the two lines intersect.
Worked examples
Ejemplo resuelto: Check a point
Is $(2, 3)$ a solution of the system $y = 2x - 1$ and $y = -x + 5$?
- First equation: $2 \cdot 2 - 1 = 3$. True.
- Second equation: $-2 + 5 = 3$. True.
- It makes both true.
Respuesta: Yes
Ejemplo resuelto: One out of two is not enough
Is $(1, 1)$ a solution of the system $y = 2x - 1$ and $x + y = 4$?
- First: $2 \cdot 1 - 1 = 1$. True.
- Second: $1 + 1 = 2$, not 4. False.
- The point is on the first line but not the second.
Respuesta: No
Error común
Checking only one equation. A point on one line is a solution of that equation only. Always substitute into both.