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Homework Sheet – Linear Equations and Simultaneous Equations
1.
Solve the following equations:
(a) 3x + 5 = 2
(b) 4x - 5 = 2
(e) 2x + 3 = -6
(f)
(i) 2(2x + 3) = 3(4x – 2)
(j)
x
4 =2
3
x
x
+4=
+1
2
4
2
x = 12
3
(c) 4x = 50
(d)
(g) 2(x + 3) = 10
(h) 3(x – 2) = 15
(k)
x  2 x 1
–
=2
3
4
2.
A number is multiplied by 5, and then 9 is subtracted. This gives an answer of 26. Find the number.
3.
The sum of three consecutive numbers is 219. Let the first number be x. Form and solve an
equation in x to find the numbers.
4.
The sum of four consecutive odd numbers is 376. Let the first number be x. What is the largest
of the four numbers?
5.
Use a graphical method to solve each of these pairs of simultaneous equations:
(a) x + y = 6
y=x–2
6.
Solve each of these pairs of simultaneous equations, making your method clear:
(a) 3x + 2y = 8
2x – y = 3
7.
(b) 3x + 4y = 24
2y = x + 2
(b) 5x + y = 16
y = 3x
(c) x + 4y = 32
x = 2y – 4
(d) x2 + y2 = 20
x = 2y
A straight line has the equation y = 5x + 3
A curve has the equation y = x2 + 2x + 3
Solve these simultaneous equations to find the points of intersection of the line and the curve.
Linear Equations and Simultaneous Equations
Q
1(a)
1(b)
1(c)
Answer
1(e)
1(g)
1(h)
1(i)
Comments
M1
x = -1
A1
4x = 2 + 5
M1
x=7÷4
Alternative methods (e.g. flow
diagrams) are acceptable.
M1
or 1.75
3
x=1
4
A1
x = 50 ÷ 4
M1
1
2
A1
x = 12 × 3 ÷ 2
M1
x = 18
A1
2x = -6 – 3
M1
x = -9 ÷2
M1
1
x= 4
2
1(f)
Mark
3x = 2 – 3
x = 12
1(d)
Answers and Mark Scheme
Students should be encouraged
to show their working.
or 12.5
delete sentences
or -4.5
A1
x
=-2+4=2
3
M1
x=2x3
M1
x=6
A1
x + 3 = 10 ÷ 2 = 5
M1
x=5–3
M1
x=2
A1
x - 2 = 15 ÷ 3 = 5
M1
or 3x – 6 = 15
x=5+2
M1
3x = 15 + 6
x=7
A1
4x + 6 = 12x – 6
M1
Multiply out brackets
8x = 12
M1
Gather like terms
x = 12 ÷ 8
M1
x=1
1
2
A1
or 2x + 6 = 10
2x = 10 – 6
or 1.5
Q
1(j)
1(k)
2.
3.
4.
5(a)
Answer
Mark
Comments
2x + 16 = x + 4
M1
Multiply throughout by 4
2x – x = 4 – 16
M1
Gather like terms
x = -12
A1
4(x + 2) – 3(x- 1) = 24
M1
Multiply throughout by 12
(4x + 8) – (3x – 3) = 24
M1
Multiply out brackets
x +11 = 24
M1
Gather like terms
x = 24 – 11
M1
x = 13
A1
5x – 9 = 26
M1
5x = 26 + 9 = 35
M1
x = 35 ÷ 5 = 7
M1
The number is 7
A1
x + (x + 1) + (x + 2) = 219
M1
3x + 3 = 219
M1
3x = 219 – 3 = 216
M1
x = 216 ÷ 3 = 72
A1
The numbers are 72, 73 & 74
A1
x + (x + 2) + (x+ 4) + (x+ 6) = 376
M1
4x + 12 = 376
M1
4x = 376 – 12 = 364
M1
x = 364 ÷ 4
M1
x = 91
A1
The largest of the four numbers is 97
A1
Set of axes, at least 0≤x≤6, 0≤y≤6, fully
labelled
B1
x + y = 6 drawn correctly
M1
y = x – 2 drawn correctly
M1
x=4 &y=2
A1 A1
Form equation
Students should be encouraged
to give answers in words to
questions in words.
Students should be encouraged
to give answers in words to
questions in words.
Q
5(b)
Answer
Mark
Set of axes, at least 0≤x≤8, 0≤y≤6, fully
labelled
B1
3x + 4y = 24 drawn correctly
M1
2y = x + 2
M1
drawn correctly
x=4 &y=3
6(a)
A1 A1
3x + 2y = 8
(x2)
-> 6x +4y = 16
2x – y = 3
( x3)
-> 6x – 3y = 9
Subtract
6(b)
6(c)
7y = 7
Selecting appropriate multiples
of equations
M1
A1
-> x = 2
A1
Subtracting
M1 Appropriate substitution
5x + y = 16
y = 3x -> 5x + 3x = 16 -> 8x = 16
M2
-> x = 2
A1
-> y = 6
A1
M1 Solving linear equation
M1 Appropriate substitution
x + 4y = 32
M2
->
y=6
A1
->
x=8
A1
x2+ y2 = 20
M1 Solving linear equation
M1 Appropriate substitution
M1 Solving linear equation
-> 4y2 + y2 = 20 -> 5y2 = 20
x = 2y
->
y2= 4 -> y = +2 or y = -2
-> x = +4 or x = -4
i.e. x = 4 & y = 2
7.
M1
-> y = 1
x = 2y – 4 -> 2y – 4 + 4y = 32 -> 6y = 36
6(d)
Comments
or
x = -4 & y = -2
M2
A2
A2
x2 + 2x + 3 = 5x + 3
M1
x – 3x = 0
M1
x(x – 3) = 0
M1
x = 0 or x = 3
A1
points of intersection are (0, 3) and (3, 18)
A1
2
Solutions must be correctly
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