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Simultaneous Equations
To solve simultaneous equations from scratch, you've got to get rid of either x or y first — to leave you with an equation with just one unknown in it. Q1 Solve the following simultaneous equations:
a) 3x = y x —y = —4
b) 2y = x y + x = 24
c) y = 4x x—y= 12
Q2 Find the value of x and y for each of the following rectangles, by first writing down a pair of simultaneous equations and then solving them.
a)
4 y + x
4
6
2 x
b)
y + x
4
11
Q3 Isobel is buying sweets. She weighs out 20 jellies and 30 toffees, which come to 230 g. She takes one of each off the scales before they get bagged up, and the weight drops to 221 g. How much does an individual toffee weigh?
Q4
Use the linear equation (the one with no x2s in it) to find an expression for y. Then substitute it into the quadratic equation (the one with x2s in it), to solve these equations:
a) y = x2 + 2 y = x + 14 b) y = x2— 8 y = 3x + 10
c) y = 2x2 y = x + 3 d) x+ 5y = 30 -5- 4 X2 + - =y
e) y = 1 — 13x y = 4x2 + 4 f) y = 3(x2 + 3) 14x + y = 1
Solve the following simultaneous equations: a) 4x + 6y = 16 y = x2— 2 c) 2 — 2y = 5 x + 2y = 5 y = 3x + 8 12y+x-2 =0
g) y2 + x2 = 20 y = 3x + 2 h) y2 + x2= 9 y + 2x = 3
d) x+y=+(y—x) x + y = 2
Two farmers are buying livestock at a market. Farmer Ed buys 6 sheep and 5 pigs for £430 and Farmer Jacob buys 4 sheep and 10 pigs for £500. a) If sheep cost fx and pigs cost fy, write down the two purchases as a pair of simultaneous equations. b) Solve for x and y.
Two customers enter a shop to buy milk and cornflakes. Mrs Smith buys 5 pints of milk and 2 boxes of cornflakes and spends £3.44. Mr Brown buys 4 pints of milk and 3 boxes of cornflakes and receives f6.03 change after paying with a £10 note. Write down a pair of simultaneous equations and solve them to find the price in pence of a pint of milk (m) and a box of cornflakes (c).
3(x5 — y) Solve = x — 3y = x — 6.
SECTION THREE — ALGEBRA