Systems of Linear Equations in Two Variables Pattern - Structured Systems

This pattern gives you a system of two equations and asks for the value of an expression (like $s + t$ or $6a$) rather than asking you to solve for each variable individually. The trick is to combine the equations directly to produce the target expression, avoiding unnecessary work.

 

Add or Subtract the Equations Directly

When the target expression can be obtained by adding or subtracting the two equations as they are, do it in one step.

$5m + 2n = 15$
$3m + 2n = 7$

If $(m, n)$ is the solution, what is the value of $2m$?

A) $22$
B) $4$
C) $-2.5$
D) $8$

Subtract the second equation from the first: $(5m + 2n) - (3m + 2n) = 15 - 7$, which gives $2m = 8$. The answer is D. No need to find $m$ or $n$ separately.

$\dfrac{s}{2} + 3t = 4$

$\dfrac{s}{2} - 3t = -10$

What is the value of $s$?

A) $-6$
B) $-3$
C) $14$
D) $\dfrac{7}{3}$

Add the two equations: $\left(\dfrac{s}{2} + 3t\right) + \left(\dfrac{s}{2} - 3t\right) = 4 + (-10)$, giving $s = -6$. The answer is A.

$3a - 5b = 10$
$3a + 5b = 2$

If $(a, b)$ is the solution, what is the value of $6a$?

A) $2$
B) $8$
C) $12$
D) $-0.8$

Add the equations: $(3a - 5b) + (3a + 5b) = 10 + 2$, giving $6a = 12$. The answer is C.

$4p + 7 = 9 + 3q$
$2p - 7 = 1 - 3q$

What is the value of $6p$?

A) $\dfrac{5}{3}$
B) $10$
C) $8$
D) $\dfrac{14}{9}$

Add the two equations: $(4p + 7) + (2p - 7) = (9 + 3q) + (1 - 3q)$. The left side gives $6p$ and the right side gives $10$, so $6p = 10$. The answer is B. Notice the $7$'s and $3q$'s both cancel.

 

Multiply First, Then Combine

When a direct add/subtract doesn't produce the target, multiply one or both equations by a constant first.

$2x + y = 7$
$-4x + 3y = 6$

The solution is $(x, y)$. What is the value of $2x + 6y$?

A) $1.5$
B) $27$
C) $4$
D) $13$

Multiply the first equation by 3: $6x + 3y = 21$. Add this to the second equation: $(6x + 3y) + (-4x + 3y) = 21 + 6$, giving $2x + 6y = 27$. The answer is B.

$5s + 7t = -22$
$2s + 3t = -9$

What is the value of $s + t$?

A) $-3$
B) $-13$
C) $-4$
D) $-1$

Subtract the second from the first: $(5s + 7t) - (2s + 3t) = -22 - (-9)$, giving $3s + 4t = -13$. That's not the target. Instead, multiply the second equation by 2: $4s + 6t = -18$. Subtract from the first: $(5s + 7t) - (4s + 6t) = -22 - (-18)$, giving $s + t = -4$. The answer is C.

$2a + 5b = 20$
$4a - 3b = 14$

What is the value of $14a - 4b$?

A) $5$
B) $2$
C) $34$
D) $62$

Multiply the first equation by 3: $6a + 15b = 60$. Add to the second: $(6a + 15b) + (4a - 3b) = 60 + 14$, giving $10a + 12b = 74$. That doesn't match either. Instead, notice that $14a - 4b$ could come from adding multiples. Multiply the second equation by 3: $12a - 9b = 42$. Multiply the first by 1: $2a + 5b = 20$. Add: $14a - 4b = 62$. The answer is D.

Tip: Before you start solving, compare the target expression's coefficients to those in the given equations. Ask yourself: "What combination of the two equations produces these coefficients?" This saves time.

 

Grouped Expressions (Substitution Shortcut)

The hardest questions wrap the variables inside expressions like $(2x - 7)$ and $(3y + 1)$ that repeat in both equations. Treat each grouped expression as a single variable.

$4(2x - 7) - 3(3y + 1) = 80$
$(2x - 7) + 3(3y + 1) = 20$

What is the value of $3(2x - 7)$?

Let $A = 2x - 7$ and $B = 3y + 1$. The system becomes $4A - 3B = 80$ and $A + 3B = 20$. Add: $5A = 100$, so $A = 20$. Then $3A = 3(2x - 7) = 60$.

$3(a + 5) + 2(b - 3) = 50$
$5(a + 5) + 2(b - 3) = 90$

What is the value of $5(a + 5)$?

Let $P = a + 5$ and $Q = b - 3$. The system becomes $3P + 2Q = 50$ and $5P + 2Q = 90$. Subtract the first from the second: $2P = 40$, so $P = 20$. Then $5P = 5(a + 5) = 100$.

$-2(p - 1) + 5(q + 6) = -15$
$2(p - 1) + 3(q + 6) = 47$

What is the value of $10(q + 6)$?

Let $A = p - 1$ and $B = q + 6$. The system becomes $-2A + 5B = -15$ and $2A + 3B = 47$. Add: $8B = 32$, so $B = 4$. Then $10B = 10(q + 6) = 40$.

$3(u + v) + 5(u - v) = 21$
$-8(u + v) - 5(u - v) = 4$

What is the value of $15(u + v)$?

Let $P = u + v$ and $Q = u - v$. The system becomes $3P + 5Q = 21$ and $-8P - 5Q = 4$. Add: $-5P = 25$, so $P = -5$. Then $15P = 15(u + v) = -75$.

Key insight: When you see the same expression repeated across both equations, don't expand it — replace it with a single letter. This turns a complex-looking system into a simple one you can solve by elimination in seconds.