QUANTITATIVE REASONING · GMAT-Q-SYS

Systems of Linear Equations

GMAT lesson development preview · DRAFT · Original learning and assessment material

Concepts and problem-solving strategy

Model the relationship

a x + b y = c, d x + ey = f

Identify constraints and signs

(2 x + 3 y) + (4 x - 3 y) = 6 x

Verify with substitution

12 x + 8 y = 400, x + y = 40 \Rightarrow 4 x = 80

Think Like a GMAT Test-Taker

Define variables with their units and translate each independent condition into an equation. Decide whether elimination can produce the requested quantity directly.

Faster Methods

Multiply equations to align one coefficient, then subtract.

Check both the total count and the total value in mixture or ticket problems.

In two-price problems, use an all-cheaper baseline. Divide the excess total value by the price difference to count the more expensive units.

Worked example 1

A venue sells 40 tickets at $12 or $8 for $400. How many $12 tickets were sold?

x + y = 40, 12 x + 8 y = 400; 4 x = 80; x = 20

Subtract eight times the count equation from the revenue equation.

Worked example 2

If 2x+3y=19 and 4x−3y=5, find x+y.

6 x = 24 \Rightarrow x = 4; y = 11/3; x + y = 23/3

Adding eliminates y. Substitution then gives the requested sum.

Worked example 3

Two equations are x+2y=10 and 3x+6y=30. Is x uniquely determined?

3 (x + 2 y) = 30; (x, y) = (10, 0), (8, 1)

The second equation repeats the first. Two valid pairs with different x values prove that x is not unique.

Worked example 4

A supplier packs 28 boxes using only small boxes holding 6 units and large boxes holding 10 units. The shipment contains 216 units. How many large boxes are there?

S + L 6 S + 10 L 6 (S + L) + 4 L 168 + 4 L = 28 = 216 = 216 = 216 \Rightarrow L = 12

Imagine that every box is small: that accounts for 168 units. Each large box adds four units beyond that baseline. The extra 48 units therefore require 12 large boxes. Check the other 16 boxes: 16×6+12×10=216.

Common Traps

A count equation and a money equation have different units.
Two proportional equations do not uniquely determine two unknowns.