Pair of Linear Equations: Graphical Method

Worked example: forming equations from a word problem

Aftab tells his daughter: "Seven years ago, I was seven times as old as you were then. Three years from now, I'll be three times as old as you'll be." Represent this algebraically.

Solution: Let Aftab's present age be $x$and his daughter's be $y$.

Seven years ago: $x-7 = 7(y-7)$

Three years from now: $x+3 = 3(y+3)$

Each condition becomes one linear equation in $x$ and $y$, turning the two equations into two lines and plotting them is exactly the graphical method.

Reading a pair of lines without plotting them

For $a_1x+b_1y+c_1=0$ and $a_2x+b_2y+c_2=0$, comparing the ratios of coefficients tells you how the two lines relate, with no graph needed:

• $\dfrac{a_1}{a_2} \ne \dfrac{b_1}{b_2}$: the lines intersect at exactly one point (a unique solution).
• $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$: the lines are coincident (infinitely many solutions).
• $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \ne \dfrac{c_1}{c_2}$: the lines are parallel (no solution).

Worked example: intersecting lines

Do $5x-4y+8=0$ and $7x+6y-9=0$ intersect, or are they parallel or coincident?

Solution: $a_1=5, b_1=-4$ and $a_2=7, b_2=6$:

Since $\tfrac{5}{7} \ne -\tfrac{2}{3}$, the lines intersect at exactly one point. This pair has a unique solution.

Worked example: coincident lines

Do $2x+3y-6=0$ and $4x+6y-12=0$ intersect, or are they parallel or coincident?

Solution: The second equation is exactly twice the first, so every ratio matches:

All three ratios are equal, so the two equations describe the same line: coincident, with infinitely many solutions.