Two Lines: Intersecting, Parallel or Coincident

Two lines, three sto­ries#

Two lin­ear equa­tions in $x$ and $y$ are really two straight lines. Solv­ing the pair means find­ing where the lines meet, and two lines can meet once, never, or every­where. First we'll see all three cases on a graph, and then learn a quick coef­fi­cient test that tells you which case you're in with­out draw­ing any­thing at all.

A pair of lin­ear equa­tions#

The gen­eral form is

$$a_1 x + b_1 y + c_1 = 0, \qquad a_2 x + b_2 y + c_2 = 0$$

A solu­tion is a pair of val­ues $(x, y)$ that sat­is­fies both equa­tions at the same time. On a graph, it's a point lying on both lines.

The three pic­tures#

Inter­sect­ing lines, exactly one solu­tion.

Take $x + y = 5$ and $x - y = 1$. Two points are enough to draw each line: the first passes through $(0, 5)$ and $(5, 0)$, the sec­ond through $(1, 0)$ and $(3, 2)$. They cross at $(3, 2)$. Check: $3 + 2 = 5$ ✓ and $3 - 2 = 1$ ✓. The pair is con­sis­tent with a unique solu­tion.

Par­al­lel lines, no solu­tion.

Now take $2x + y = 4$ and $2x + y = 8$. They have the same slope but dif­fer­ent inter­cepts: the first runs through $(0, 4)$ and $(2, 0)$, the sec­ond through $(0, 8)$ and $(4, 0)$. Like two rail­way tracks, they never meet. And that makes sense, since no pair $(x, y)$ can make $2x + y$ equal to both $4$ and $8$. The pair is incon­sis­tent.

Coin­ci­dent lines, infi­nitely many solu­tions.

Finally take $x - 2y = 3$ and $3x - 6y = 9$. Look closely: the sec­ond is just the first mul­ti­plied by $3$, so they are the same line drawn twice. Every point on it, $(3, 0)$, $(5, 1)$, $(1, -1)$, and infi­nitely many more, is a solu­tion. The pair is depen­dent (and there­fore also con­sis­tent).

The coef­fi­cient test#

Draw­ing takes time, and read­ing a cross­ing point off graph paper is unre­li­able when the coor­di­nates are frac­tions. A faster way is to com­pare ratios of coef­fi­cients:

One cau­tion before you use it: put the equa­tions in the form $ax + by + c = 0$ before you read off $a$, $b$, $c$, and that includes the sign of $c$. Read­ing $c$ from $2x + y = 4$ as $+4$ instead of $-4$ is the most com­mon slip of all.

Prac­tice#

1. Solve graph­i­cally: $x + 2y = 6$ and $x - y = 0$. Ver­ify the solu­tion alge­braically.
2. Clas­sify with­out draw­ing: (a) $5x - 3y + 1 = 0$ and $10x - 6y + 2 = 0$; (b) $x + 3y = 4$ and $2x + 6y = 5$; (c) $2x - y = 1$ and $x + y = 5$.
3. For what value of $k$ does the pair $3x + ky = 6$ and $6x + 4y = 12$ have infi­nitely many solu­tions?
4. For what value of $k$ does the pair $kx + 2y = 5$ and $3x + y = 1$ have no solu­tion?
5. Two lines are par­al­lel and both pass through $(1, 2)$. What can you con­clude?

Answers#