Two Simultaneous Linear Equations

What you’ll learn

Find a pair that satisfies two equations at once.

រកគូតម្លៃដែលបំពេញសមីការពីរព្រមគ្នា។

Meaning and use

A solution to two simultaneous equations must satisfy both. Elimination combines equations to remove one unknown; substitution replaces one unknown using the other equation. Some pairs have no solution or infinitely many, so do not assume every pair has a unique intersection.

ចម្លើយប្រព័ន្ធសមីការត្រូវបំពេញសមីការទាំងពីរ។ វិធីបំបាត់អថេរ ឬជំនួសជួយរកគូតម្លៃ។ ប្រព័ន្ធខ្លះអាចគ្មានចម្លើយ ឬមានច្រើនអនន្ត។

Formulas

Two-equation system

\[ax+by=c,\qquad dx+ey=f\]

A unique solution condition

\[ae-bd\ne0\]

Symbols and units

Symbol	Meaning	Unit or restriction
x, y	Unknown values	According to the problem
a, b, c, d, e, f	Known coefficients and constants	Compatible units within each equation
ae − bd	Determinant of the coefficient pair	Nonzero for a unique solution

Conditions and limits

• Any scaling step must multiply every term in that equation.
• If elimination gives 0 = nonzero, there is no solution; 0 = 0 needs further interpretation.
• ចម្លើយចុងក្រោយត្រូវពិនិត្យទាំងពីរសមីការ។

Worked examples

Elimination by addition

Solve \(x+y=10\) and \(x-y=2\).

1. Add equations: \(2x=12\).
2. Find \(x=6\).
3. Substitute into first: \(6+y=10\), so \(y=4\).

x = 6, y = 4

បូកសមីការទាំងពីរ ដើម្បីបំបាត់ y។

A second system

Solve \(2x+y=11\) and \(x+y=7\).

1. Subtract second from first: \(x=4\).
2. Substitute \(4+y=7\).
3. Find \(y=3\).

x = 4, y = 3

ដកសមីការដោយធ្វើលើគ្រប់តួទាំងពីរខាង។

Common mistakes

Instead of	Use	Why
A pair is a solution if it satisfies one equation.	It must satisfy both.	A system imposes both conditions.
Multiply only the x term when scaling an equation.	Multiply every term on both sides.	Otherwise the equation changes its solution set.

Practice

Solve these before revealing the worked solutions. Include units where relevant.

សាកល្បងដោះស្រាយមុនបើកមើលចម្លើយ។ សូមបង្ហាញជំហាន និងឯកតាដែលពាក់ព័ន្ធ។

1. Solve x + y = 8 and x − y = 2.
2. Check x = 2, y = 1 in 2x + y = 5 and x + y = 3.
3. Do x + y = 3 and x + y = 5 have a common solution?

Answers and steps

Show solutions / បង្ហាញដំណោះស្រាយ

1. x = 5, y = 3

   Add to get 2x = 10, then substitute.

2. The pair is correct.

   Both left sides match their right sides.

3. No.

   The same sum cannot equal two different numbers.

Quick recap

• A solution satisfies every equation.
• Scale whole equations.
• Check whether the system has a unique solution.