Substitution and elimination

Why solve with algebra

In the course on graphs and consistency, you drew a pair of lines and read the crossing point from the graph. A graph is only as exact as your ruler. If the answer is $x = \dfrac{7}{3}$, a graph cannot show it. Algebra gives the exact answer every time.

A pair from a fair

At a school fair, Amit buys $2$ bags of peanuts and $1$ cup of juice for ₹$35$. Neha buys $3$ bags of peanuts and $2$ cups of juice for ₹$60$. What does each item cost?

Let a bag of peanuts cost ₹$x$ and a cup of juice cost ₹$y$. The two bills give a pair of equations:

The substitution method

Substitution writes one letter in terms of the other, and then puts that into the second equation.

1. Choose the easier equation and write one letter in terms of the other. From the first equation, $y = 35 - 2x$.
2. Substitute it into the other equation, so that only one letter is left: $3x + 2(35 - 2x) = 60$.
3. Solve for that letter: $3x + 70 - 4x = 60$, so $-x = -10$ and $x = 10$.
4. Put it back to find the other letter: $y = 35 - 2 \times 10 = 15$.
5. Check in both equations. $2 \times 10 + 15 = 35$ and $3 \times 10 + 2 \times 15 = 60$. Both hold.

A bag of peanuts costs ₹$10$ and a cup of juice costs ₹$15$.

The elimination method

Elimination makes the coefficients of one letter equal, or opposite, in the two equations. Then adding or subtracting removes that letter. Two facts make this allowed:

• You may multiply both sides of an equation by the same number that is not zero.
• You may add or subtract two equations, because equal amounts added to equal amounts stay equal.

Take the fair pair again, $2x + y = 35$ and $3x + 2y = 60$.

1. Match one letter. Multiply the first equation by $2$. It becomes $4x + 2y = 70$.
2. Subtract the second equation: $(4x + 2y) - (3x + 2y) = 70 - 60$, which gives $x = 10$.
3. Find the other letter by putting $x = 10$ into either equation. The answer is $y = 15$.

Elimination gives the same answer as substitution, as it must.

When the coefficients are opposite

Sometimes no multiplying is needed. In $3x + 2y = 17$ and $5x - 2y = 7$, the terms in $y$ are $+2y$ and $-2y$. Add the equations:

Then $3 \times 3 + 2y = 17$ gives $y = 4$.

When neither letter matches

Take $2x + 3y = 12$ and $3x + 5y = 19$. No coefficient matches, so make them match.

1. Choose a letter to remove, say $x$. Its coefficients are $2$ and $3$, and the smallest number both divide is $6$.
2. Multiply the first equation by $3$ and the second by $2$: $6x + 9y = 36$ and $6x + 10y = 38$.
3. Subtract: $y = 2$.
4. Put it back: $2x + 6 = 12$, so $x = 3$.

Which method to use

• Use substitution when a letter already has the coefficient $1$ or $-1$, as in $y = 35 - 2x$.
• Use elimination when two coefficients match, are opposite, or can be matched by a small multiplier.
• Always put your answer back into both of the original equations.

When both letters disappear

Sometimes elimination removes both letters. Take $x + 2y = 3$ and $2x + 4y = 9$. Double the first equation to get $2x + 4y = 6$, and subtract the second:

That is false for every $x$ and $y$, so the pair has no solution. The lines are parallel. If you reached $0 = 0$, the equations would describe the same line, with infinitely many solutions. The conditions on the coefficients in the graphs course tell the same story.

Practise with new numbers

Try it in Python

The program solves the fair pair by substitution and by elimination, and checks that the two answers agree.

Common mistakes

• Multiplying only one side. If you multiply $2x + 3y = 7$ by $4$, the result is $8x + 12y = 28$. The $7$ must be multiplied too.
• Subtracting with a lost sign. In $(4x + 2y) - (3x + 2y)$, every term of the second bracket changes sign.
• Stopping at one letter. The pair has two unknowns, so the answer needs both values.
• Skipping the check. A slip in step $2$ or $3$ shows up at once when you put the answer back.

Remember these

• Substitution: write one letter in terms of the other, put it into the second equation, solve, and put back.
• Elimination: match the coefficients of one letter, add or subtract, solve, and put back.
• If both letters disappear, a false statement means no solution and $0 = 0$ means infinitely many.
• Check the answer in both equations.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Solve $y = 2x$ and $3x + y = 15$ by substitution.

2. Solve $4x + 5y = 34$ and $4x - 5y = 14$.

3. Solve $x + 3y = 11$ and $2x - y = 1$ by any method.

4. After elimination, a pair gives $0 = 8$. What does this say about the two lines?

5. Ravi multiplies $2x + 3y = 7$ by $4$ and writes $8x + 12y = 7$. What is his mistake?