Pair of Linear Equations in Two VariablesGraphical Method of Solution of a Pair of Linear Equations

Akhila's two lines met at exactly one point. Is that the only thing two straight lines can do? Picture drawing two lines on the same sheet of paper. There are not many possibilities, so let's hunt for all of them.

The blue line is fixed: . The three numbers in the equation build the green line , which starts out as Akhila's Equation (2).

Drag the three numbers to complete all three missions.

${a} + ${b} − ${c}

Tip

The lines keep the same tilt whenever is exactly −2 times — just like the blue line's own numbers, and .

Mission complete! So two lines on a flat page can do exactly three things. They can (i) cross at a single point, and then the pair has exactly solution. They can (ii) run parallel and never meet, and then the pair has solution. Or they can (iii) coincide, one lying on top of the other, so every point of the line is shared and the pair has solutions.

Consistent, Inconsistent, Dependent

Each of those three fates has a name.

A pair of equations with at least one solution is called consistent. A pair with no solution at all is called inconsistent. And a pair whose two equations are really the same line in disguise, one just a multiple of the other, is called dependent. Such equivalent equations share every point of their line, so a dependent pair has infinitely many solutions, which means a dependent pair is always consistent too.

Drag a line from each name to what it promises:

A consistent pair

An inconsistent pair

A dependent pair

Has no solution at all

Has at least one solution

Is one line written twice, so every point on it is a solution

Putting names to pictures: crossing lines mean a pair with a unique solution; parallel lines mean an pair; and coincident lines mean a pair, which is also consistent, with infinitely many solutions.

The Ratio Test

The line tamer showed you all three pictures, but drawing is slow. Could we predict the picture straight from the numbers, without drawing at all? Take three pairs of equations, write each pair in general form, and compare the ratios , and of matching coefficients:

Table 3.1

Sl No.	Pair of lines	Graphical representation	Algebraic interpretation
1.	,	Intersecting lines	Exactly one solution (unique)
2.	,	Coincident lines	Infinitely many solutions
3.	,	Parallel lines	No solution

Read along each row and a pattern jumps out. For lines and , when the lines are (i) intersecting, ; (ii) coincident, ; (iii) parallel, .

The converse is true as well, for every pair of lines: the ratios alone tell you the picture. Try a few pairs of your own and check.

For a pair of lines and :

→ the lines intersect → exactly one solution (consistent).
→ the lines are parallel → no solution (inconsistent).
→ the lines are coincident → infinitely many solutions (dependent).

Why the ratios decide everything

If all three ratios are equal, the second equation is just the first one multiplied by a number, the same line in disguise, so the two lines . If only holds and the constant ratio is different, the lines have the same tilt but sit in different places, so they run parallel and . And if even the first two ratios differ, the tilts differ, so the lines must .

Common trap: don't stop after checking and . When those two are equal, it is the third ratio that decides between parallel, with , and coincident, with .

Predict, Then Look

Here are the three pairs from Table 3.1 again. Before you see any graph, use the ratio test to predict what each pair of lines does. Compare and first; only if they are equal does matter. Each graph draws itself as soon as you answer.

(i) and : the lines

(ii) and : the lines

(iii) and : the lines are

Table 3.1 fits into three boxes. Without looking back at it, drag each ratio test, and each pair of equations, into the picture it produces:

a₁/a₂ ≠ b₁/b₂

a₁/a₂ = b₁/b₂ = c₁/c₂

a₁/a₂ = b₁/b₂ ≠ c₁/c₂

x − 2y = 0 and 3x + 4y − 20 = 0

x + 2y − 4 = 0 and 2x + 4y − 12 = 0

2x + 3y − 9 = 0 and 4x + 6y − 18 = 0

Intersecting lines

Parallel lines

Coincident lines

Ratio Detective

Now put the ratio test to work at speed. You get a fresh pair of equations each time. Compare the ratios in your head and call the picture before anyone draws it.

Speed round: get five right to move on.

${e1}

${e2}

Solving by Drawing

Time to use the graphical method for real.

Example 1: Check graphically whether the pair of equations

(1)

(2)

is consistent. If so, solve them graphically.

Solution: A straight line is fixed by just of its points, so we find two solutions of each equation. They are listed in Table 3.2.

Table 3.2

x	0	6
	2	0

x	0	3
	−4	−2

Now build Fig. 3.1 yourself: plot A (0, 2), B (6, 0), P (0, −4) and Q (3, −2) by tapping where each one belongs.

Joining the points gives the lines AB and PQ of Fig. 3.1. Both lines pass through B (6, 0), so the pair has the solution and , which means the pair is . The ratio test agrees: , so the lines had to cross somewhere.

Example 2: Graphically, find whether the following pair of equations has no solution, unique solution or infinitely many solutions:

(1)

(2)

Solution: The fractions in Equation (2) look messy, but watch what happens when we multiply it through by : it turns into , which is Equation (1) exactly. So both equations describe one and the same line. The lines are , and the pair has .

You could plot a few points to check this. Here is that check, live.

The blue line is Equation (1). The dashed green line is , which is Equation (2) rescaled so that its -coefficient is (at it is Equation (2) itself). Change and watch the coefficients. Does the line ever move? At which value of does it become Equation (1) letter for letter?

${a}

Example 3: Champa went to a sale to buy some pants and skirts. When her friends asked how many of each she had bought, she said, "The number of skirts is two less than twice the number of pants I bought. It is also four less than four times the number of pants." Help her friends find how many pants and skirts Champa bought.

Solution: Let be the number of pants and the number of skirts. Champa's two sentences become:

(1)

(2)

Two solutions of each equation are listed in Table 3.3.

Table 3.3

x	2	0
	2	−2

x	0	1
	−4	0

Build Fig. 3.2 beside the table: plot the four points A, B, P and Q, and keep an eye on which one ends up on both lines.

The lines of Fig. 3.2 cross at (, ). So solves the pair: Champa bought pair of pants and did not buy any .

Always check an answer against the story: twice 1, less 2, is 0 skirts, and four times 1, less 4, is also 0 skirts. Both of Champa's sentences hold.

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