Theory question 102
Linear and Simultaneous Equations(a) Solve the simultaneous equations: 6y + 5x = −12; 4y − 3x = 11 (b) in the diagram below, ADC is a striaght line /CD/ = 48cm, /BD/ = 36cm and /AD/ = ycm. Find the value of y.
Show the model answer
Model answer
(a) Multiply the first equation by 3: 18y + 15x = −36. Multiply the second by 5: 20y − 15x = 55.
Add: 38y = 19, so y = ½.
Substitute: 4(½) − 3x = 11, so −3x = 9 and x = −3.
Answer: x = −3, y = ½.
(b) BD is perpendicular to AC and the marked angles are equal (<ABD = <BCD), so triangles ABD and BCD are similar.
AD/BD = BD/DC, so y/36 = 36/48.
y = 36 × 36 ÷ 48 = 27 cm.
(Alternatively: tan <BCD = 36/48 = 0.75 and tan <ABD = y/36 = 0.75, so y = 27 cm.)
Explanation
In (b), the angle marked at B equals the angle at C, which makes the two smaller right-angled triangles similar.