Algebraic Method — Substitution

Why Algebraic Methods?

Graphs are great for seeing the solution, but they're slow and imprecise — especially when the answer isn't a nice whole number. Algebraic methods give the exact solution quickly. The first is the substitution method.

The idea: From one equation, express one variable in terms of the other. Then substitute that expression into the second equation, turning it into a single equation in one variable, which you can solve easily.

Think of it this way: we use one equation to 'replace' a variable in the other, reducing two unknowns to one.

[Board Important] Substitution is best when one variable already has coefficient 1 (or is easy to isolate), e.g. $x + 2y = 5$ where $x = 5 - 2y$ is immediate.

The Substitution Steps

Let's solve $x + 2y = -1$ and $2x - 3y = 12$.

1. Isolate a variable. From the first equation, $x = -1 - 2y$.
2. Substitute into the second: $2(-1 - 2y) - 3y = 12$.
3. Solve the one-variable equation: $-2 - 4y - 3y = 12 \Rightarrow -7y = 14 \Rightarrow y = -2$.
4. Back-substitute to find the other variable: $x = -1 - 2(-2) = -1 + 4 = 3$.

So the solution is $x = 3$, $y = -2$.

Key Point: After finding one variable, always substitute back into the simplest equation to find the other.

[Board Important] Verify your answer by plugging $(3, -2)$ into both original equations. This catches arithmetic slips and earns confidence (and sometimes a mark).

What If There's No Solution / Infinitely Many?

Substitution also reveals the special cases automatically.

• If you end up with a false statement like $0 = 5$, the variable cancels and the equation is impossible — the pair has no solution (parallel lines).
• If you end up with a true statement like $0 = 0$, the equation is always true — the pair has infinitely many solutions (coincident lines).

Example (no solution)

$x + y = 3$ and $2x + 2y = 9$. From the first, $x = 3 - y$. Substitute: $2(3 - y) + 2y = 9 \Rightarrow 6 = 9$, which is false. So no solution.

Key Point: A contradiction ($0 =$ non-zero) ⇒ no solution. An identity ($0 = 0$) ⇒ infinitely many solutions.

[Board Important] Don't panic if the variables vanish — read the leftover statement. False ⇒ no solution; true ⇒ infinitely many.

Substitution in Word Problems

The real power of substitution shows in word problems. The recipe:

1. Define the two variables clearly.
2. Form the two equations from the conditions.
3. Solve by substitution.
4. Interpret the answer in the context (with units), and check it makes sense.

Worked outline

'The sum of two numbers is 20 and one is 4 more than the other.'

• Let the numbers be $x$ and $y$. Then $x + y = 20$ and $x = y + 4$.
• Substitute: $(y + 4) + y = 20 \Rightarrow 2y = 16 \Rightarrow y = 8$, so $x = 12$.

Key Point: When one condition directly gives one variable in terms of the other (like $x = y + 4$), substitution is the fastest method.

[Board Important] Always end a word problem by stating the answer in words: 'The two numbers are 12 and 8.' Bare values can lose a presentation mark.

Solved Examples

Example 1: Basic substitution

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

Solution:

1. From the first: $x = 14 - y$.
2. Substitute: $(14 - y) - y = 4 \Rightarrow 14 - 2y = 4 \Rightarrow y = 5$.
3. Then $x = 14 - 5 = 9$.

Final Answer: $x = 9$, $y = 5$.

Takeaway: Isolate, substitute, solve, back-substitute.

Example 2: Substitution with fractions avoided

Solve: $2x + 3y = 11$ and $x - 2y = -12$.

Solution:

1. From the second (coefficient 1 on $x$): $x = 2y - 12$.
2. Substitute: $2(2y - 12) + 3y = 11 \Rightarrow 4y - 24 + 3y = 11 \Rightarrow 7y = 35 \Rightarrow y = 5$.
3. $x = 2(5) - 12 = -2$.

Final Answer: $x = -2$, $y = 5$.

Takeaway: Isolate the variable with coefficient 1 to avoid fractions.

Example 3: No solution case

Solve: $x + y = 3$ and $2x + 2y = 9$.

Solution:

1. From the first: $x = 3 - y$.
2. Substitute: $2(3 - y) + 2y = 9 \Rightarrow 6 - 2y + 2y = 9 \Rightarrow 6 = 9$ (false).
3. The variables cancelled to give a false statement.

Final Answer: No solution (inconsistent).

Takeaway: A false numeric statement ⇒ no solution.

Example 4: Infinitely many solutions

Solve: $x - 2y = 5$ and $2x - 4y = 10$.

Solution:

1. From the first: $x = 5 + 2y$.
2. Substitute: $2(5 + 2y) - 4y = 10 \Rightarrow 10 + 4y - 4y = 10 \Rightarrow 10 = 10$ (true).

Final Answer: Infinitely many solutions.

Takeaway: A true statement like $10 = 10$ ⇒ infinitely many solutions.

Example 5: Two numbers word problem

The sum of two numbers is 20 and their difference is 4. Find them.

Solution:

1. Let the numbers be $x$ and $y$: $x + y = 20$, $x - y = 4$.
2. From the second: $x = y + 4$. Substitute: $(y+4) + y = 20 \Rightarrow 2y = 16 \Rightarrow y = 8$.
3. $x = 12$.

Final Answer: The numbers are 12 and 8.

Takeaway: State the answer in words at the end.

Example 6: Coin/age type problem

A father is three times as old as his son. After 12 years, he will be twice as old. Find their present ages.

Solution:

1. Let the father be $x$ and son be $y$: $x = 3y$.
2. After 12 years: $x + 12 = 2(y + 12)$.
3. Substitute $x = 3y$: $3y + 12 = 2y + 24 \Rightarrow y = 12$, so $x = 36$.

Final Answer: Father 36 years, son 12 years.

Takeaway: Substitute the direct relation ($x = 3y$) into the other equation.

Example 7: Solve and verify

Solve $3x - y = 3$ and $9x - 3y = 9$, and state the type.

Solution:

1. From the first: $y = 3x - 3$.
2. Substitute: $9x - 3(3x - 3) = 9 \Rightarrow 9x - 9x + 9 = 9 \Rightarrow 9 = 9$ (true).

Final Answer: Infinitely many solutions (the equations are equivalent).

Takeaway: The second equation is just 3× the first — they are the same line.

Example 8: Fractional answer

Solve: $2x + y = 7$ and $3x - 2y = 0$.

Solution:

1. From the first: $y = 7 - 2x$.
2. Substitute: $3x - 2(7 - 2x) = 0 \Rightarrow 3x - 14 + 4x = 0 \Rightarrow 7x = 14 \Rightarrow x = 2$.
3. $y = 7 - 4 = 3$.

Final Answer: $x = 2$, $y = 3$.

Takeaway: Substitution works smoothly even when the second equation has no constant.

Example 9: Speed of boat (current) word problem

A boat goes 30 km downstream and 30 km upstream. If still-water speed is $x$ and current speed is $y$, and downstream speed is $x + y = 15$ km/h while upstream speed is $x - y = 9$ km/h, find $x$ and $y$.

Solution:

1. From the second: $x = 9 + y$.
2. Substitute: $(9 + y) + y = 15 \Rightarrow 2y = 6 \Rightarrow y = 3$.
3. $x = 12$.

Final Answer: Still-water speed 12 km/h, current 3 km/h.

Takeaway: Downstream = $x + y$, upstream = $x - y$; solve by substitution.

Example 10: Two-digit number

The sum of the digits of a two-digit number is 9. If 9 is added to the number, the digits are reversed. Find the number.

Solution:

1. Let tens digit $= x$, units $= y$. Then $x + y = 9$ and $(10x + y) + 9 = 10y + x$, i.e. $9x - 9y = -9 \Rightarrow x - y = -1$.
2. From $x + y = 9$ and $x - y = -1$: substitute $x = y - 1$ into the first: $(y-1) + y = 9 \Rightarrow y = 5$, $x = 4$.
3. Number $= 10(4) + 5 = 45$.

Final Answer: The number is 45.

Takeaway: Translate 'digits reversed' carefully into $10y + x$.