Simultaneous Equations

Two equations, two unknowns, and three ways to find the pair that satisfies both — plus what it means when there is no such pair.

หลักสูตรแกนกลาง / สสวท.: จำนวนและพีชคณิต (Number and Algebra)ตรงกับ ม.3 บทที่ 6 ระบบสมการเชิงเส้นสองตัวแปร
Common Core (US): 8.EE, A-REI
Cambridge: Lower Secondary Stage 9 — Algebra: expressions, equations and formulae

6.1 Elimination

The method

1Write both equations in the form $a x + b y = c$
2Multiply one or both so that one variable has matching or opposite coefficients
3Add if the signs are opposite, subtract if they match
4Solve for the remaining variable, then substitute back
5Check the pair in BOTH original equations

ตัวอย่าง 1

Solve $2 x + 3 y = 12$ and $4 x - 3 y = 6$

วิธีทำ

1.The $y$ coefficients are $+ 3$ and $- 3$ — opposite, so add
2. $6 x = 18$ , giving $x = 3$
3.Substitute: $6 + 3 y = 12$ , so $y = 2$
4.Check in the second: $12 - 6 = 6$

ตอบ $x = 3$ , $y = 2$

ตัวอย่าง 2

Solve $3 x + 2 y = 16$ and $5 x - 4 y = - 10$

วิธีทำ

1.Multiply the first by $2$ so the $y$ terms become $+ 4$ and $- 4$ : $6 x + 4 y = 32$
2.Add: $11 x = 22$ , so $x = 2$
3.Substitute: $6 + 2 y = 16$ , so $y = 5$

ตอบ $x = 2$ , $y = 5$

ข้อควรระวัง

When you multiply an equation, multiply EVERY term including the one on the right of the equals sign. Missing it changes the equation into a different one, and the answer comes out wrong with no sign that anything happened.

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