Technique

Add and subtract when the coefficients are reversed

Algebra • Equations • Simultaneous Equations

Statement

For

a x + b y = p

and

b x + a y = q

, adding gives

(a + b) (x + y) = p + q

and subtracting gives

(a - b) (x - y) = p - q

. Each combination yields

x + y

x - y

in one line, and halving their sum and difference then gives

x

and

y

without ever multiplying an equation through.

How to use it

1Look for it whenever the two equations use the same two coefficients in swapped positions, however large those coefficients are — the point is to avoid multiplying by them.
2Signs travel with the coefficients: for $a x - b y = p$ and $b x - a y = q$ the sum gives $(a - b) (x + y)$ and the difference gives $(a + b) (x - y)$ , so read off which combination each produces rather than assuming.
3When the question asks only for $x + y$ , $x - y$ or $x y$ , one of the two combinations may already be the answer and neither $x$ nor $y$ need be found.

Worked example

Solve

97 x + 89 y = 283

and

89 x + 97 y = 275

Step-by-step

1
$186 (x + y) = 558 \Rightarrow x + y = 3$
Why: Adding the equations: both coefficients become $97 + 89 = 186$ .
2
$8 (x - y) = 8 \Rightarrow x - y = 1$
Why: Subtracting: both coefficients become $97 - 89 = 8$ .
3
$x = 2, y = 1$
Why: Half the sum and half the difference of $x + y$ and $x - y$ .

Answer

x = 2

y = 1

Watch out

It needs the coefficients genuinely reversed; a pair such as

3 x + 5 y

and

4 x + 5 y

gains nothing from adding. The subtraction step divides by

a - b

, so it says nothing when

a = b

— and then the two equations are either the same line or parallel.

Also known as

add and subtract the equations, symmetric coefficients, reversed coefficients, elimination by addition and subtraction, x+y and x-y from a linear pair

Add and subtract when the coefficients are reversed

Statement

Worked example

Appears in past papers

2020 (II)

2020 (I)

2018 (I)