Graphs and Consistency of Pairs of Equations: Quick Revision

A brief note on Graphs and Consistency of Pairs of Equations. Read it before a test. Try each question at the end before you open its answer.

The graph of a pair

• A pair is $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$.
• Each equation draws a straight line. A solution of the pair is a point on both lines.
• Two points draw a line. Use a third as a check. Always check a solution in both equations.

Three cases

• Lines cross: exactly one solution. The pair is consistent.
• Parallel lines: no solution. The pair is inconsistent.
• Coincident lines: infinitely many solutions. Consistent and dependent.
• A pair can never have exactly two solutions.

The ratio test

• $\dfrac{a_1}{a_2} \neq \dfrac{b_1}{b_2}$: one solution.
• $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2}$: no solution.
• $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$: infinitely many solutions.
• Write both equations as $ax + by + c = 0$ first. To find $k$, set the ratios equal and check the third.

A worked example

For which $k$ does $2x + 3y - 7 = 0$ and $4x + ky - 9 = 0$ have no solution?

1. Equal first two ratios. $\dfrac{2}{4} = \dfrac{3}{k}$, so $k = 6$.
2. Check the third ratio. $\dfrac{-7}{-9} = \dfrac{7}{9}$ is not $\dfrac{1}{2}$.
3. Decide. The lines are parallel, so $k = 6$ gives no solution.

Practice with new numbers

A program to try

The program decides the number of solutions without dividing, by cross-multiplying.

Mistakes to avoid

• Saying "no solution" for coincident lines. They have infinitely many.
• Stopping at two ratios. Equal first two ratios mean parallel or coincident.
• Forgetting the standard form. The signs of the constants change the third ratio.
• Checking only one equation. A solution must fit both.

Check yourself

1. Do $x + y = 6$ and $2x + 2y = 12$ have a solution?

2. Classify $x - 2y - 3 = 0$ and $2x - 4y - 5 = 0$.

3. A pair has parallel lines. What is the pair called?