Intersection of two lines

The intersection of two lines is the point that lies on both of them. There, the $y$ computed from one equation and the $y$ computed from the other agree exactly. So to find it we set the two expressions for $y$ equal and collapse them into a single equation. That is precisely what solving a system of equations means¹.

Solving the system

Find where the two lines $y = 2x - 1$ and $y = -x + 5$ meet. At the meeting point the two $y$ values are equal.

\begin{align*} 2x - 1 &= -x + 5 \\ 3x &= 6 \\ x &= 2 \end{align*}

Substituting back, $y = 2 \cdot 2 - 1 = 3$ . So the intersection is $(2, 3)$ .

Let us check $(2, 3)$ in the other equation as well: $-2 + 5 = 3$ , so the point really does lie on both lines. Since an intersection is a point satisfying both equations at once, this substitution is all the verification the answer needs.

Number of intersections

The key idea is that the number of intersections equals the number of solutions of the equation. Looking for the intersection of $y = a_1 x + b_1$ and $y = a_2 x + b_2$ leads to the following.

(a_1 - a_2)x = b_2 - b_1

Whether this linear equation can be solved decides how the lines meet.

Condition	The equation	The lines
$a_1 \neq a_2$	one solution	cross at exactly one point
$a_1 = a_2$ , $b_1 \neq b_2$	no solution	are parallel and never meet
$a_1 = a_2$ , $b_1 = b_2$	every $x$ works	coincide

In other words, as long as the slopes differ, two lines must cross at exactly one point. Two non-parallel lines failing to meet is impossible. Here the slopes are $2$ and $-1$ , so there is exactly one intersection.

Another form of the equations

The lines are not always handed to you as $y = \cdots$ . They may arrive in the form $ax + by = c$ , and then it is easier to add or subtract the equations to eliminate a variable.

\begin{align*} 2x - y &= 1 \\ x + y &= 5 \end{align*}

Adding these two cancels the $y$ terms and leaves $3x = 6$ , giving $x = 2$ once again. Solving for $y$ and equating, or eliminating directly, lands on the same intersection.

Applications

The crossing of a demand line and a supply line is the market equilibrium
The crossing of two cost lines is the break-even point where the cheaper plan changes
Solving a system of two linear equations in two unknowns

The large dot on the graph is the intersection $(2, 3)$ of this example.

System of linear equations, Wikipedia