Simultaneous Equations

Simultaneous Equations

Simultaneous equations are two (or more) equations that are both true at the same time. Solving them finds the values of x and y that satisfy both equations.

Linear Simultaneous Equations

Two methods: elimination and substitution.

Elimination

Make the coefficients of one variable the same, then add or subtract to eliminate it.

Worked Example: Solve 3x + 2y = 16 and 5x + 2y = 22.

• The y coefficients are already the same (both 2y)
• Subtract equation 1 from equation 2: (5x + 2y) − (3x + 2y) = 22 − 16
• 2x = 6 → x = 3
• Substitute into equation 1: 3(3) + 2y = 16 → 2y = 7 → y = 3.5
• Check in equation 2: 5(3) + 2(3.5) = 15 + 7 = 22 ✓

Worked Example: Solve 2x + 3y = 12 and 5x − 4y = 1.

• Multiply equation 1 by 4: 8x + 12y = 48
• Multiply equation 2 by 3: 15x − 12y = 3
• Add: 23x = 51 → x = 51/23 — this is messy, so let's try the other variable
• Multiply equation 1 by 5: 10x + 15y = 60
• Multiply equation 2 by 2: 10x − 8y = 2
• Subtract: 23y = 58 → y = 58/23 — still messy! Let me recheck...
• Actually: 23y = 58, y = 58/23. Hmm, let's use neater numbers.

Worked Example (corrected): Solve 2x + 3y = 11 and 3x − y = 8.

• Multiply equation 2 by 3: 9x − 3y = 24
• Add to equation 1: 2x + 3y + 9x − 3y = 11 + 24
• 11x = 35 → x = 35/11... Let me use: 3x + 2y = 12 and x − 2y = 0
• Add: 4x = 12 → x = 3
• From equation 2: 3 − 2y = 0 → y = 1.5
• Check: 3(3) + 2(1.5) = 9 + 3 = 12 ✓

Substitution

Rearrange one equation to make a variable the subject, then substitute into the other.

Worked Example: Solve y = 2x + 1 and 3x + 2y = 16.

• Substitute y = 2x + 1 into the second equation: 3x + 2(2x + 1) = 16
• 3x + 4x + 2 = 16
• 7x = 14 → x = 2
• y = 2(2) + 1 = 5

Linear-Quadratic Simultaneous Equations (Higher)

One equation is linear, the other contains x², y² or xy. Always use substitution.

Worked Example: Solve y = x + 3 and x² + y² = 29.

• Substitute y = x + 3 into the quadratic: x² + (x + 3)² = 29
• x² + x² + 6x + 9 = 29
• 2x² + 6x − 20 = 0
• x² + 3x − 10 = 0
• (x + 5)(x − 2) = 0
• x = −5 → y = −2, or x = 2 → y = 5
• Solutions: (−5, −2) and (2, 5)

Worked Example: Solve y = 3x − 1 and y = x² + x − 3.

• Set equal: x² + x − 3 = 3x − 1
• x² − 2x − 2 = 0
• Using the formula: x = (2 ± √(4 + 8)) / 2 = (2 ± √12) / 2 = 1 ± √3
• x = 1 + √3, y = 2 + 3√3 or x = 1 − √3, y = 2 − 3√3

Graphical Interpretation

The solutions of simultaneous equations are the coordinates where the two graphs intersect. A linear and quadratic pair can have 0, 1 or 2 intersections.

Exam Tips

• Always give answers as pairs — state both x and y for each solution
• For elimination, remember: same signs subtract, different signs add
• For linear-quadratic, you must expand the bracket carefully — the most common error is forgetting to square both terms in (x + 3)²
• Always check your answers by substituting back into both original equations