Solving a Special Type of Pair of Linear Equations

This lesson shows a fast way to solve a special pair of linear equations: one where the coefficient of $x$ in the first equation equals the coefficient of $y$ in the second, and vice versa. Instead of the usual elimination, we add and subtract the equations to get much smaller numbers.

The given pair

We are asked to solve:

$23x + 37y = 32 \quad (1)$

$37x + 23y = 88 \quad (2)$

Notice the pattern: the coefficient of $x$ in $(1)$ is $23$, which is also the coefficient of $y$ in $(2)$. Likewise $37$ is the coefficient of $y$ in $(1)$ and of $x$ in $(2)$. Making coefficients match in the normal way would create large numbers, so we use a shortcut.

Step 1: Add the two equations

Adding $(1)$ and $(2)$, the $x$ terms give $23 + 37 = 60$ and the $y$ terms give $37 + 23 = 60$:

$60x + 60y = 120$

Divide every term by $60$:

$x + y = 2 \quad (3)$

Step 2: Subtract the two equations

Subtracting $(2)$ from $(1)$:

$(23 - 37)x + (37 - 23)y = 32 - 88$

$-14x + 14y = -56$

Divide every term by $14$:

$-x + y = -4 \quad (4)$

Step 3: Solve the simpler pair

Equations $(3)$ and $(4)$ have tiny coefficients, so they are easy to solve. Add them:

$(x + y) + (-x + y) = 2 + (-4)$

$2y = -2 \quad\Rightarrow\quad y = -1$

Step 4: Find x

Put $y = -1$ into equation $(3)$:

$x + (-1) = 2 \quad\Rightarrow\quad x = 3$

Check. In $(1)$: $23(3) + 37(-1) = 69 - 37 = 32$. In $(2)$: $37(3) + 23(-1) = 111 - 23 = 88$. Both hold, so the solution is correct.

$x = 3, \quad y = -1$

Key takeaways

• When the $x$ and $y$ coefficients are swapped between two equations, adding and subtracting them is faster than ordinary elimination.
• Adding gives one simple equation and subtracting gives another, each divisible by a common factor.
• Solving these two small equations and substituting back gives $x = 3$ and $y = -1$, which checks out in both original equations.