SSC CGL Maths · Linear Equations

Solving Linear Systems

Remove one unknown at a time, by adding, subtracting or substituting equations, until one equation in one unknown is left; then check the answer in an equation you did not use.

All Linear Equations notes

Why this matters

Ten questions in the bank, six EASY and four MODERATE: three solve a pair of equations, five solve a set of three, and two are fastest by putting the options back into the equations. Every word problem and age problem later in this chapter ends with an equation or a pair of equations solved the way this page teaches.

Concept 1 of 3: Solving two equations in two unknowns

One equation in x and y is true for endless pairs of numbers. A second equation keeps only the pairs that fit both, and for most pairs of equations that is a single pair. To find it, get rid of one unknown so that one equation in one unknown is left.

Definition

Elimination: multiply one or both equations so that one unknown has the same coefficient in both, then add or subtract the equations to remove it.
Substitution: when one equation already gives one unknown on its own, such as $y = 2x + 3$ , put that expression into the other equation.
Two equations of the form y = ...: set the two right sides equal, since both equal $y$ .
Sum and difference: if $x + y = S$ and $x - y = D$ , add to get $2x = S + D$ and subtract to get $2y = S - D$ .
Find the second unknown from either equation, then check the pair in the other one. The check takes five seconds and catches most slips.

Sum and difference of two numbers

x + y = S,\; x - y = D \;\Rightarrow\; x = \frac{S + D}{2},\; y = \frac{S - D}{2}

Sthe sum of the two numbers
Dthe difference, larger minus smaller

Worked example

Solve

2x + 3y = 13

and

4x - y = 5

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 15 Sep 2025, Shift 2 · Q70Easy

Example 1 · Linear Equations · Solving Linear Systems

Solve system:

y = 2x + 1

and

y = -x + 4

Stopping at the first unknown

Finding

x

is half the job. An option may print only the other number, or a pair with the right

x

and a wrong

y

. Find both values and check the pair in the equation you did not use last.

Subtracting only part of an equation

When you subtract one equation from another, subtract every term, the constant included:

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

, not

2y

, and the right sides are subtracted too.

Drill 2 more on solving two equations in two unknowns

Concept 2 of 3: Solving three equations in three unknowns

Three unknowns need three equations. Use two different pairs of them to remove the same unknown, and you are back to two equations in two unknowns, which you already know how to solve.

Definition

Pick the unknown that is easiest to remove: one with coefficient 1, or one that is missing from an equation already.
Combine two different pairs of equations to remove that same unknown. This gives two equations in the other two unknowns.
Solve that pair, then put both values into the simplest original equation to get the third unknown.
Fast route, sign pattern: when the equations differ only in signs, such as $x + y + z$ and $x + y - z$ , subtracting one from the other leaves $2z$ at once.
Fast route, symmetric set: for $x + y = p$ , $y + z = q$ , $z + x = r$ , add all three. Each unknown appears twice, so $x + y + z = \frac{p + q + r}{2}$ ; subtract each equation from this to get the unknown it leaves out.
If the options print full values, testing them can be quicker still (the next concept).

Symmetric set of three equations

x + y = p,\; y + z = q,\; z + x = r \;\Rightarrow\; x + y + z = \frac{p + q + r}{2}

p, q, rthe three pair sums

Worked example

Solve

x + y + z = 9

x - y + 2z = 7

and

2x + y - z = 3

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 11 Sep 2024, Shift 1 · Q66Easy

Example 2 · Linear Equations · Solving Linear Systems

Solve the given set of three equations to get the value of the unknowns:

x + 2z = 3

x + 2y + 3z = 5

3x - 5z = -13

Removing a different unknown from each pair

If one pair of equations loses

x

and the other pair loses

y

, you still have all three unknowns spread over two equations. Remove the same unknown from both pairs.

Using the same pair twice

Two combinations of the same two equations give only one new fact. Each of the two pairs must bring in the third equation, or the system cannot be finished.

Drill 4 more on solving three equations in three unknowns

Concept 3 of 3: Testing the options in a system of equations

A solution has to satisfy every equation. With four answers printed, putting them back into the equations is often faster than solving, and the last check is built in.

Definition

Choose the equation with the smallest numbers and test every option in it. It usually rules out two or three options.
Put the survivor into every other equation. A wrong option can pass one equation and fail the next.
When the question says a given point solves the system and asks for a constant, substitute the point: each equation becomes an equation in the constant alone.
When each option names a constant and a solution together ("if a = 2, then x = 3, y = 1"), put the constant into the equations first, then test the pair.
When it breaks: if the options print only a combination (a sum, a ratio, one unknown), testing cannot check the rest of the solution. Solve instead.

Worked example

Which of these solves

2x + y - z = 1

x - y + 3z = 8

and

3x + 2y + z = 10

? (A)

(1, 2, 3)

(B)

(2, 1, 3)

(C)

(1, 3, 2)

(D)

(3, 2, 1)

, each written as

(x, y, z)

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 12 Sep 2024, Shift 3 · Q64Moderate

Example 3 · Linear Equations · Solving Linear Systems

Find the values of x, y and z, so as to satisfy the equations given below:

5x - 3y + 7z = 22

;

3x - 5y - 2z = -46

;

2x - 2y + 5z = 24

Accepting an option after one equation

An option that satisfies one equation is not yet a solution. Options are often built to pass the first equation, so check the survivor in every equation of the system.

Testing the pair before fixing the constant

When an option says "if a = 6, then x = 4, y = 2", the equations change with

a

. Put the value of

a

in first, then test the pair in the new equations.

Drill 1 more on testing the options in a system of equations

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Solving two equations in two unknowns
Sum and difference of two numbers
$x + y = S,\; x - y = D \;\Rightarrow\; x = \frac{S + D}{2},\; y = \frac{S - D}{2}$
Solving three equations in three unknowns
Symmetric set of three equations
$x + y = p,\; y + z = q,\; z + x = r \;\Rightarrow\; x + y + z = \frac{p + q + r}{2}$

Watch out for (6)

Stopping at the first unknown
→ Solving two equations in two unknowns
Subtracting only part of an equation
→ Solving two equations in two unknowns
Removing a different unknown from each pair
→ Solving three equations in three unknowns
Using the same pair twice
→ Solving three equations in three unknowns
Accepting an option after one equation
→ Testing the options in a system of equations
Testing the pair before fixing the constant
→ Testing the options in a system of equations

Test yourself on Linear Equations

20 past SSC CGL questions from this chapter, timed at 12 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Linear Equations chapter test Drill all 10 questions