Solving Pairs of Equations Algebraically: Quick Revision

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

The three methods

• Substitution: write one letter in terms of the other, put it into the second equation, solve, and put back.
• Elimination: multiply to match the coefficients of one letter, add or subtract, solve, and put back.
• Cross-multiplication: write both equations as $ax + by + c = 0$. Then $x = \dfrac{X}{D}$ and $y = \dfrac{Y}{D}$, where $X = b_1c_2 - b_2c_1$, $Y = c_1a_2 - c_2a_1$ and $D = a_1b_2 - a_2b_1$.
• Always check the answer in both equations.

When both letters disappear

• A false statement such as $0 = 5$ means no solution: the lines are parallel.
• The statement $0 = 0$ means infinitely many solutions: the two equations give the same line.
• In cross-multiplication, $D = 0$ means the same thing, so the formula cannot be used.

Equations in 1 over x and 1 over y

• Let $u = \dfrac{1}{x}$ and $v = \dfrac{1}{y}$, solve for $u$ and $v$, then take reciprocals.
• With $x + y$ and $x - y$ in the denominators, let $u = \dfrac{1}{x + y}$ and $v = \dfrac{1}{x - y}$.
• The answer must give $x$ and $y$, not $u$ and $v$.

Word problems

• Name the unknowns with their units, and write one equation for each fact.
• A two-digit number with tens digit $x$ and units digit $y$ is $10x + y$.
• Downstream speed is $u + v$, and upstream speed is $u - v$.
• Check the answer against the words of the problem.

A worked example

Solve $7x - 2y = 16$ and $3x + 4y = 36$.

1. Match $y$. Multiply the first equation by $2$: $14x - 4y = 32$.
2. Add the second equation: $17x = 68$, so $x = 4$.
3. Find $y$. $3 \times 4 + 4y = 36$ gives $y = 6$.
4. Check. $7 \times 4 - 2 \times 6 = 16$ and $3 \times 4 + 4 \times 6 = 36$.

Practice with new numbers

A program to try

The program solves three pairs with the formulas, and checks each answer in both equations.

Mistakes to avoid

• Multiplying only one side of an equation.
• Losing a sign when you subtract one equation from another.
• Using the wrong sign for $c$ in cross-multiplication.
• Stopping at $u$ and $v$ instead of going back to $x$ and $y$.

Check yourself

1. Solve $x + y = 14$ and $x - y = 4$.

2. Write $2x - 3y = 8$ in the form $ax + by + c = 0$. What is $c$?

3. Solve $\dfrac{3}{x} + \dfrac{4}{y} = 5$ and $\dfrac{6}{x} - \dfrac{4}{y} = 4$.

4. A boat goes $15$ km/h downstream and $9$ km/h upstream. What is the speed of the stream?