Source; File A-Level Cambridge Principles of Mathematics 10
File: ‘Principles of Mathematics 10’
(14/6/2017)
Objective: The team of authors is targeting to enable
students to develop kernel of reflexive solution model which grants breaking
the problem at axiomatic level classification which may be consumed recursively
in similar intellect by understanding reasons’ splits. The acumen so developed
will be long lasting for applicability of solution mindset dimensions (*).
(*): Whether ‘Brain-Power’ Or
‘Heart-Impulse’ Plinths ‘Human Gen’? http://b4gen.blogspot.ca/2017/03/whether-brain-power-or-heart-impulse.html
Applicability Domain is natural language
diversification semantics into precise algebraic expression, say as;
Atomic fact is a ‘phrase’: freshman terminology
group of words commonly used to constitute sentence/ clause. However, to cut it
in short to start with; phrases and liner expressions that means an algebraic
equation of one or more variable(s) with the linear power.
Problem solving: perceive the problem, model the problem and
apply modular solution of the pertinent reflexive form, a case;
Modelling Linear Relations: Translating
phrases ß à Linear
expression
Specimen worksheet; one variable, one
operation, one numeric value;
1).
Sum of x and 2 x + 2
2). t divided
by 8
t / 8
3). Product of
9 and y
9 * y
4). Subtract 8 from z
z – 8
5). Two- fifth of h (2/ 5) * h
6). 5 multiplied by b 5 * d
7). One- third added to k k + 1/3
8). c decreased
by 7
c – 7
9).
One-half of a
(1/2) * a
TIPS for Solution: Identify arithmetic operation symbols of [plus or add
(+)], [minus or subtract or decreased (–)],
[multiplication or product (*)], [division or ratio (/)], variable and
numeric value.
Insert arithmetic operation between variable and numeric value.
Pages
13 – 14
23 – 24
35 – 36
49 – 50
63 – 64
75 – 76
87 – 88
Use Mathematical
Symbols on: https://www.mathsisfun.com/symbols.html
|
Pages
01 – 02
03 – 04
05 – 06
07 – 08
09 – 10
11 – 12
13 – 14
15 – 16
17 – 18
19 – 20
21 – 22
23 – 24
25 – 26
27 – 28
29 – 30
31 – 32
33 – 34
35 – 36
37 – 38
39 – 40
41 – 42
43 – 44
45 – 46
47 – 48
49 – 50
51 – 52
53 – 54
55 – 56
57 – 58
59 – 60
61 – 62
63 – 64
65 – 66
67 – 68
69 – 70
71 – 72
73 – 74
75 – 76
77 – 78
79 – 80
81 – 82
83 – 84
85 – 86
87 – 88
89 – 90
91 – 92
93 – 94
95 – 96
97 – 98
|
Chapter I Linear Systems
1.1 Connect English With Mathematics and Graphing
Lines
1.2 The
method of Substitution
1.3
Investigate Equivalent Linear Relations and Systems
1.4
The Method of Elimination
1.5
Solve Problems Using Linear Systems
Chapter I
Review
Chapter
2 Analytical Geometry
2.1
Midpoint of a Line Segment
2.2 Length of a Line Segment
2.3 Apply Slope, Mid Point and Length Formulas
2.4
Equation for a Circle
Chapter 2 Review
Chapter 3 Geometric Properties
3.1
Investigate Properties of Triangles
3.2
Verify Properties of Triangles
3.3 Investigate
Properties of Quadrilaterals
3.4
Verify Properties of Quadrilaterals
3.5
Properties of Circles
Chapter 3 Review
Chapter
4 Quadratic Relations
4.1
Investigate Non Linear Relations
4.2
Quadratic Relations
4.3
Investigate Transformation of Quadratics
4.4 Graph y = a(x – h)2 + k
4.5
Quadratic Relations of the Form y = a (x – r) (x – s)
4.6
Negative and Zero Exponents
Chapter 4 Review
Chapter
5 Quadratic Expressions
5.1
Multiply Polynomials
5.2
Special Products
5.3
Common Factors
5.4 Factor
Quadratic Expression x2 + b x + c
5.5
Factor Quadratic Expression ax2 + b x + c
5.6
Factor of a Perfect Square Trinomial & Diff. of Sqr
Chapter 5 Review
Chapter
6 Quadratic Equations
6.1 Maxima
and Minima
6.2
Solve Quadratic Equations
6.3
Graph Quadratic Equations
6.4
The Quadratic Formula
6.5
Solve Problems Using Quadratic Equations
Chapter 6 Review
Chapter
7 Trigonometry of Right Triangles
7.1
Investigate Property of Similar Triangles
7.2
Use Similar Triangles to Solve Problems
7.3
The Tangent Ratio
7.4
The Sine and Cosine Ratios
7.5 Solve
Problems Involving Right Triangles
Chapter 7 Review
Chapter
8 Trigonometry of Right Triangles
8.1
The Sine Law
8.2
The Cosine Law
8.3
Find Angles Using The Cosine Law
8.4
Solving Problems Using Trigonometry
Chapter 8 Review
Enrichment
Questions
****
1.1
Connect English With Mathematics and Graphing Lines. [Using Tips of solution given above, resolved as under]
A
1.
Translate
each phrase into an algebraic expression.
ANS
a).
five more than twice a number. à 2x + 5 OR 2*x + 5
b).
three less than twice a number. à (1/4) x– 3 OR (1/4)* x–
3 OR compound fraction
(numerator / denominator)
c).
the product of a number and another number increased by 7. à m (n + 7) OR m* (n +
7)
d).
a value decreased by the fraction one half.
à n – ½ OR n
– (1/2)
2.Translate each phrase into an algebraic
expression. ANS
a).
four times a length. à 4l
b).
triple a distance. à 3d
c).
thirty percent of a number. à 0.30n
d).
six percent of a price. à 0.06p
3.
For
each of the following, write an opposite meaning word or phrase. ANS
a).
decreased. à increased
b).
subtracted. à added
c).
less than. à more than
d).
minus.
à plus
e).
added to. à subtracted from
f).
more than. à less
than
4.
Translate
each sentence into an algebraic equation.
ANS
a).
One sixth of a number increased by 15, is 42. à (1/60 n + 15 =
42
b).
Three times a number, decreased by 4, is 5 more than 6 times number. à 3n – 4 = 6n + 5
c).
When tickets to a soccer game cost $4 each, collected the revenue $320.à 4p = 320
d).
The total length of the base and height of a triangle 15 cm. à b + h = 15
B
TIPS of Solution on Intersection of Lines, the common point where
two lines cross, which has the same
(x, y) coordinates, on Line 1 and
Line 2, that equivalence resolves y and then x.
5.
Find
the point of intersection for each pair of lines by graphing. Checking your
answer. ANS
a.
y
= 2x + 5, y = x +1 è Solution:
2x + 5 = x +1 è x = –4 è by substitution y = –3 à (x,
y) is (–4, –3)
b.
y
= –2x + 3, y = 3x +8 è Solution:
–2x + 3 = 3x + 8 è x = –1 by substitution y = 5 à (x,
y) is (–1,5)
c.
y
= 4x + 8, y = ½ x +1 è Solution:
4x + 8 = ½ x +1 è x = –2 by substitution y = 0 à (x,
y) is (–2,0)
d.
y
= 4/5 x +2, y = ¾ x + 3 è Solution:
4/5 x +2 = ¾ x +3 èx = 20 by substitution y =18 à (x,
y) is (20,18)
6.
Find
the point of intersection for each pair of lines by graphing. Checking your
answer.
ANS
a.
x
+ y = 6, 2x –y = 6 è Solution: y = 6 –x & y = 2x –6 è 6 –x = 2x –6 èx = –4 è by substitution y = –3 à (x, y) is (–4, –3)
3x + 4y = 6, 2x – 4y = è Solution:
4y = 6 –3x & 4y = 4 –2x è 6 –3x = 4 –2x è x =2 by substitution y = 0 à (x, y) is (2,
0)
b.
x
– y = 3, 3x + y = 5 è Solution:
y = x –3 & y = 5–3x è x –3= 5 –3x è 4x = 8 è x =2 by substitution y= –1à (x, y) is (2, –1)
c.
2x
+3y =5, x – 3y = 4 è Solution:
by elimination 3x= 9 è x= 3 by substitution y = –1/3
à (x, y) is (3, –1/3)
7.
Use
Technology Use Graphing calculator or The Geometer’s Sketchpad ® to find the
point
of intersection for
each pair of lines, where necessary round answers to the nearest hundredth. ANS
a.
y
= 8x +5, y = –7x –6 è Solution: 8x +5 = –7x –6 è 15x = –11 è x = .73 è by substitution y= –7*(–11/15)
–6 is –0.87à (x, y) is (0.73, –0.87)
b.
y
= –3x +5, y = 4x +7 è Solution: –3x +5 = 4x +7 è 7x = –2 è x = –0.29 è by substitution y = (–3 * –0.29)
+5 is 5.87 à (x, y) is (-0.29, 5.86)
c.
y
= 2.3x +9, y = 4.5x –10 è Solution:
2.3x +9 = 4.5x –10 è 2.2x = 19 è x =8.64 by substitution y= 2.3*8.64
+9 is 28.88 à (x, y) is (8.64, 28.87)
d.
y
= –0.3x + 2.4, y = –0.2x +3.5 è Solution:
–0.3x +2.4 = –0.2x + 3.5 è x = –11 by substitution y = –0.3*(-11)
+ 2.4 is 5.7 à(x, y)
is (-11, 5.7)
8.
Sarah
deliver flyers in the summary to make some extra money. She charges $10.00 per
hour.
Ads–R–Us
Delivery Service charges $120 for the season.
ANS
a).
Write an equation for the amount Sarah charges to deliver flyers for the
season. Equation of amount to seasonal deliver is C
=10x
b). Write an equation for the amount
Ads–R–Us. Delivery services charges. Given reason is C
= $120 à 120 = 10x à x = 12
c). What is the intersection pointing
to of the two linear equations? Intersecting point: C=
120 & x =12 à (x, y) is (12, 120)
d). In the context of this question,
what does the point of intersection represent? Sarah charges same price of work as Aimee
9.
Use
Technology Savio works for a cellular phone company. He is paid $90 per day
plus $2.00 for each cellular
phone sale. Aimee also
works for the cell phone company, but she makes $120 each day. ANS
a). Write an equation to represent the amount
that Savio earns in one day. Graph the equation. Equation Savio earnings
à E =
90 + 2n
b). Write an equation to represent the amount
that Aimee earns in one day. Graph the equation on same grid. Aimee earnings / dayà E = 120
c). How many cellular phones must Savio sell
to make as such in a day as Aimee. Sale of Savio = Aimee à 15 cellular phones
10.
Kristen
has a total $1000. She has an account 4%
interest per year and in bond of 6.5%Interest.
If she has $50 in
simple interest at year end, how much was invested rate-wise. $50
earning at 4% & 6.5% for $1000 splità 0.04x+ 0.06y =50
11.
Graph
the equations y = 2x + 1, y = –3x + 6, & y = ½x + 5/2 on same grid. Explain;
Given 3 lines intersect at point (1, 3). Graph
Saved http://www.meta-calculator.com/online/dr1xzjg2tfh4
12.a).
Can you solve the linear system y = 3x-2 and 6x –2y –4 = 0à y =
3x –2 à NO, à The 2 lines are same lines
intersect everywhere, many solutions Graph Saved NO intersection graph ???
b).
Solve the linear system y = 4x –3 and 8x –2y +5 =0 à y =
4x + 5/2 à y = 4x –3 & y = 4x + 5/2 à (x, y) is NO interesect, The 2 lines
are parallel ??? Graph
Make again http://www.meta-calculator.com/online/h084wr1dgk27
c). Explain tell
without solving, how many solution is a linear system has. à Explanation;
If the two lines have the same slope and y-intercepts à with infinite
solutions,
if they have same slope and different intercept lines à parallel, NO
solution, if different slope lines than one solution ???
MAKE EXAMPLES.
NOTE Graphic
solution for y = f(x) form only
****
1.2
The
Method of Substitution: Principles of Mathematics 10, pages 20 – 28 TIPS on Solutions by Substitution: to solve one of the equations for one variables,
and plug this into the other equation, the simplest equation used for
substitution of one variable into another equation, to form single variable
system. For instance; 4x + y = 24, and y = –4x + 24
substitute y of the second equation into the first and resolve x.
A.
1 . Sole each linear system by
substitution. Check answers.
a). y = 2x +5, x + y =
8 Solution: substitute
the simplest equation y = 2x +5 into x + y = 8 è x + 2x + 5 = 8à 3x = 3à x =1,
applying x in y = 2x +5 è y =2*1 + 5 à y = 7,
(x, y) is (1, 7)
b). y = 3x –7, x + 2y =7
Solution: substitute
the simplest equation y = 3x –7 into x +2y = 7à x +2*(3x –7) = 7à 7x = 21à x = 3
applying x in y = 3x –7 è y=3*3–7à y =2,
(x, y) is (3, 2)
c). y = –x + 3, 2x +3y=
5 Solution: substitute
the simplest equation y = –x + 3 into 2x +3y = 5 è 2x+ 3(–x + 3))= 5 è x =4
applying x in y = –x + 3 è y = –4+ 3 à y= –1,
(x, y) is (4, –1)
d). 3x + 4y= –4, x= 2 –3y
Solution: substitute
the simplest equation x = 2 –3y into 3x+ 4y= –4 è 3(2 –3y)+ 4y= –4 è y= –2/5 applying y in x= 2 –3y è x= 2 –3( –2/5) à y
=3.2, (x, y) is (–2/5, 3.5)
2.
In each pair, decide which equation one
rewrite one variable in terms of other variable.
a).
x + 3y = 4, 4x + 2y =7 Solution: choose simplest
for one variable in terms of another: x + 3y = 4
b).
2x + 5y = 8, 2x + y =6 Solution: choose simplest
for one variable in terms of another: x + 3y = 4
3.
a).
Is (1, 1) the solution for the following linear system? Explain how;
3x + 4y = 7, 2x + 5 = 8 Solution: NO, the value set (1, 1) does not satisfy 2nd
equation 2x+ 5 =8.
b).
Is (4, –3) the following of the following system. Explain;
3x – 2y =18, 2x + 3y = -1 Solution: YES, the value set (4, –3) satisfies both the
equations.
B.
4.
Solve
by substitution and Check
a)
x+
3y = 5, 4x + 2y = 10 Solution: substitute the
simplest equation x +3y= 5 into 4x + 2y = 10è y =1,
applying y in x +3y =5 è x =1,
(x, y) is (1, 1)
b). 5a + b =4, 3a + 2b = –6 Solution:
substitute the simplest equation 5a + b =4 into 3a+ 2b = –6 è a= 2,
applying a in 3a+ 2b = –6 èb = –6, (x, y)
is (2, -6)
c). x –2y =5, 2x + 3y = 17 Solution:
substitute the simplest equation x –2y =5 into 2x + 3y= 17è y =1,
applying y in 2x+ 3y = 17 è x =7,
(x, y) is (7, 1)
d). 2m –3n = –10, 4m+ n =1 Solution:
substitute the simplest equation 4m+ n =1 into 2m
–3n = –10 èm = –1/2 applying in 4m+ n =1 è n=3,
(x, y) is (-1/2, 3)
5. Find intersection points of check.
a). 5x= y +11, 2x +y =3 Solution: elimination
leads to x= 2, on substitution y = –1à intersecting point (x,
y) is (2, –1)
b). m + 3n = 4, 4m + 2n + 4 = 0 Solution:
elimination leads to n =2 on substitution m= –2 à intersecting points
(m, n) = (–2, 2)
6. Kyle reads for twice as many hours per
week as Santiago. Together they read 24 Hours a week.
a). How to assign variables? Solution:
Variant answer, Kyle and Santiago reading hours, let it be k and s respectively
b). Write equation for the first sentence Kyle reads /week,
2 times than Santiago. Solution: k = 2s
c). Write equation for the second sentence, Kyle &
Santiago read 24 hours. Solution: k + s =24
d). Find by substitution hours each person spent each
speak. Solution: k = 2s and k + s =24 à 3s = 24 à s = 8 and k =16
7. Let N represents pairs of shorts Nyiri
bought, and R of Raven Nyiri and Ravert buy total 9 nine pair of shorts. Raven
purchase 6 less than twice shorts as Nyiri.
a). Assign variable, write equation to represent Nyiri and
Raven buying. Solution:
N + R =9
b). Write an equation to represent the third sentence.
Raven 6 less than twice Raven. Solution: R = 2N –6
c). Solve linear system by substitution. Solution: N +R =9 and R = 2N –6 à N =4 & R =5
d). If the short costs $15.99 each. How much did each
spend. Solution: Nyiri spent 4 * 15.99 = $63.96 and Raven
spent 5* 15.99 = $79.99
8. Sports-mania charges $450 for the hall
and $16 per meal. Sports-To-Go charges $330 for the hall and $20 per meal;
a). Assign variables write 2 equations to represent the
info. Let C cost of rent and n meals. Solution: C = 450+
16n and C= 330 +2n
b). Solve the linear system to find number of guests
for which both halls charge the same amount. Solution:
Charges at both hall for 40 guests
9. Ron makes
comforter charging the first type $35 for material and $60/h for labour. For
the second type charges
are $105 for
the material and $25/h for labour. For what number of hours are the cost
same.
Solution: C1 = 35+ 60h, C2 =
105+ 25h à C1 = C2 à Solution: 35+ 60h = 105+ 25h à h = 2
10. Christina makes leather belts, for the
first type she charges $80 for the material and $50/h for labor. For the second
type charges $100 for the material and $40/h for labor. . For what
number of hours are the costs the same? Solution: 80 +50h = 100 +40h à h =2 of
labor
C.
11. The following lines intersect to form a
triangle; y= x +3, 2x + y =6, x +y = 7
a). Find the coordinates of each vertices. Solution: Vertex 1: y= x +3, 2x + y =6 à (1,
4), Vertex 2: y= x +3, x +y = 7 à (2,
5), Vertex 3: x+ y= 7 and 2x +y =6 à (-1,
8)
b) Is this a right-angle, Explain, check
slopes, Solution: y =x +3 and y = –x+ 7 by y =
mx +c where m is slopeà one slope is 1 and another –1à right-angle
12, Simplify each equation and solve
linear system, round by tenth;
a). 2(x +1) + 3(y +2) =
15, x + (y –1) =2, Solution:2x +2 + 3y + 6 = 15
and x +y = 3 à 2x+ 3y= 7 & x + y= 3 à (2,
1)
b). 2(x –1) + y = 5, 4x
–3(y + 2) = 15, Solution: simplify both equations &
rewrite à 2x +y =7 and 4x –3y = 21 à (4.2, –1.2)
c). 3(x –1) – (y+1) =1,
4(x + 1) + 2(y –1) = 10, Solution: simplify both
equations & rewrite à 3x –y = 5 and 4x + 2y =8 à (1.8,
0.4)
d). 2(x +1) + 3(y + 2)
= 10, – (x +3) + 2(y –1) = 1, Solution: simplify both
equations & rewrite à 2x + 3y =2 and –x +2y =6 à (-2,
2)
13. The following lines intersect
at one point;
3x +5y =10, x +2y =4,
5x + ky =10 Find the coordinates of intersection of three lines and value of k.
Solution: 3x +5y =10
and x +2y =10 à (0,
2), employing x +2y =4 and 5x + k *y= 10 à on resolving for intersection
(0, 2), k =5
***
1.2
Investigate Equivalent Linear Relations and Equivalent
Linear Systems.
A.
1.
Which two equations are equivalent?
A.
y = x + 3
B.
y = ½ x + 4
C.
2y = x + 8
2.
Which is not an equivalent linear relation?
A.
6y = 2x +4
B.
y = (1/3) x + 2/3
C.
9y = 2x + 3
D.
3y = x +2
3.
Write two equivalent equations for each.
a). y
=5x –3
b). 4x
+3y =12
c). y
= (2/3) x +5
d). x + y =7
4. Which
two of the following liners equations will have the same graph?
A. 2y
= ½ x + 4
B. y =
¼ +1
C. 4y
= x +4
B.
5. If y = 3x –7 and 3y = k*x –21 are equivalent linear equations, what
is the value k?
6. a). If y =2x –7 and 4y = k*x –32 are equivalent linear equations, what is the value of
k?
b). If y = 5x –4 and 2y = 10x +k
are equivalent linear equations, what is the value of k?
c). If 3y =4x –2 and k*y =8x –4 are
equivalent linear equations, what is the value of k?
7. The total number of males and
females in Endi’s main class is 1.4.
a). State how u will assign
variables?
b). Write an equation to represent
this situation. Then, write an equivalent linear equatiom.
8. The total number of dimes and
quarters in Marijan’s piggy bank is 82.
a). State how you will assign
variables.
b). Write an equation to represent
the situations. Then write an equivalent linear equation.
9. The perimeter of a rectangle is 1.8m.
a). State how you will assign
variables.
b). Write an equation to represent
this situation. Then, write an equivalent linear equation.
10.
A linear system is given,
4x
–8y = 10 à (1)
x + y = 5 à (2)
Explain why the following is an
equivalent linear system;
2x –4y =5 à (3)
2x + 2y =10 à (4)
11. A linear system is given;
y = (3/4) x –2 (1)
y = (2/3) x + 1 (2)
Explain why the following is an
equivalent linear system;
4y = 3x –8 (3)
3y = –2x + 3 (4)
C.
12. a). A linear system is given;
4(2x –3) + 53y +8) =4
3(4x +2) –(2y –3) = 7
Show that the following is an equivalent
linear system;
8x + 15y = –24 (1)
12x
–8y =
–11 (2)
b). y= (8/15)*x –(11/8)
Explain why the following is an
equivalent linear system.
8x –15 y = –24 (3)
12x – 8x = –11 (4)
13.a). Graph the following linear systems
on grid paper.
y =x
y –x +4
b). State the point of intersection of two
lines.
c). Show that the two lines are
perpendicular to each are.
d). We are two equivalent linear systems
to the following linear system.
y = x
y = –1 x +4
14.a). Graph the following linear
system of the two lines.
b). State the point of
intersection of 2 lines
c). Show that the two lines are
perpendicular to each other.
d). Write two linear systems
equivalent to the following linear systems.
y = 2 x
y = (–1/2)*
x +5
****
1.4 The Method of Elimination
A.
1.
Solve
using the method of elimination. Check each solution.
a)
x
+ y = 4
2x –y = 5
b)
x –y
= –2
3x + y = –10
c)
x
+ 4y = 10
x + y = 1
d) 4x + 3y= 10
2x + 3y =2
2.
Solve
using the method of elimination. Check each equation.
a)
x
+ 3y = 7
–x +2y = 3
b) 2x + 4y =10
–2x +3y = 11
c)
x
+ 5y = 7
x + 3y = 5
d) 4x + 5y = 7
4x + 2y = 1
3.
Solve
the method of elimination.
Check each equation.
a)
x
+ 5y =8
x + y = 4
b) x + 3y =3
2x + 4y =2
c)
3x
+ 4y =14
2x + y =1
d) 3x + 5y = 4
3x + 6y = 6
4.
Find
the point of intersection of each pair of lines. Check solutions.
a)
2x + 3y = 7
3x –2y
= 4
b) 2x + 4y =2
5x –3y = 5
c)
2x
+ 3y = 5
3x + 2y = 5
d) 2a + 5b =5
3a + 2b =13
B
5.
Find
the point of intersection of each pair of lines. Where necessary express
answers as fractions in lowest terms. Check solution.
a)
3x
+ 5y = 10
2x + 5y = 7
b) 3x + 6y = 9
2x –6y = 8
c)
3x
+4y = 7
5x + 3y =8
d) 3x +5y =4
2x +3y = 7
e)
4x
+ 2y 3
2x + 3y= 7
f)
2m –3n =1
4m + 2n = 3
6.
Ziadame
selling skates. A pair of hockey skates costs $58.00 and a pair of skates cost
$56.00 One shift, Ziadne sold 32 pairs of skates, on receipt $1828.
a)
How
many pairs of hockey skates did Ziadame sell?
b) How many pairs of figures
skates did Ziadane sell?
7.
Sadia
selling popcorns, large boxes for $5.00 and small boxes for $3.00. She sold 60
boxes, on receipt of $260.
a)
How
many large boxes of popcorn did Sadia sell?
b) How many small boxes did she
sell?
8.
Consider
the system
3x –4y =–1
4x +y =5
a)
Solve
by elimination
b) Solve by substitution
c)
Which method do you prefer? Why?
9.
Solve
each linear system, Check solution
a)
0.3x
–0.2y = 11
0.5x +0.4y = 55
b) 0.4a –0.2b = 20
0.3a + 0.5b = 54
10. Expand
and simplify, solve the linear system and Check
a) 3(2x –1) – (y +3) =1
4x(1 + 2x) –3(2 –y) =3
b) 4(a –2) –2(b +3) = –6
3(a –2) – 2(b –1) = 6
c)
2(m
+1) + 4(n –1) =6
3( m –1) + 4(n +1) = 7
C
11. Explain how to solve the system 5x +
4y –3 and 3x +5y = 8, by elimination, not to answer.
12. What happens when solve this system by
elimination
3x +4y =8
6x + 8y =0
13. What happens when you solve this system
by elimination?
a) 3a/ 5 – b/2 = 9
3a/4 + b/3 =17
c)
(m
–3)/ 5 + (n +2)/ 4 =1
(m +4)/ 3 + (n
–3)/2 =2
14. Solve be elimination
dx + my =c
ex + ny = 8
***
1.5
Solve Problems Using Linear Systems
A.
1.
Zara
at part-time flower shop making flowers a total 30 flowers for a customer using
two types of flowers. She is asked to use 4 types as many carnations as roses.
How many each will be in he arrangement?
2.
Daniel
has 126 Canadian and British stamps. He determines that he has 38 more Canadian
stamps than British stamps. How many of each type of stamps does he have?
3.
Faiza
and Terry held a fund-raising sale. They sold large bottles of water for $3 and
small bottles for $2, sold 180 bottles and collected $440. How many of each
size of bottles did they sale?
4.
Marley
invest the earnings of $2050. She invests the part of money at 7%/ year, and the
rest at 6%/ year. After one year these investments earn $18 simple interest.
How much did she invest at each rate?
B.
5.
Shirley
went to visit Butterfly Museum. There were butterflies in the tropical garden
area. Shirley calculated that there 18 more painted lady butterflies than
monarch butterflies. How many of each type of butterflies were in the tropical garden?
6.
Two
cartons of milk contain different percent of butterfat. How much 1 % milk needs
to be mixed with how much 5% milk to give 10 L of 4% milk?
7.
Nathaniel needs to make 30 L of 28% sulfuric
acid solution. In the Chemistry office, he finds bottles of 20% sulfuric acid and
50% sulfuric acid. What volume of each should he mix in order to make the 28%
solution?
8.
One type of chocolate mixture contains 30%
nuts by mass. A second type of chocolate mixture contains 30% nuts by mass.
What mass of each type of chocolate mixture to make 500-grams that will have
24% nuts, by mass?
9.
Use
Technology to join the Bridge club. Taha must pay an initial fee of $150 and
monthly fee of $20. If he joins the Bid Bridge club, he must pay initial fee of
$100 and monthly fee $25.
a)
After
how many months is the cost same at either Bridge club?
b) If Taha plans to join a bridge
club for eight months, which club should he join?
c)
If
Taha decides to join a bridge club for one year, which club should he join?
10.
For
a mathematics conference, Mary decides to order royal blue T-shirts for all the
volunteers. It will cost $10 shirt for large size and $8 per shirt for the
medium size. Mary orders a total of 40 T-shirts and spends $350.
a)
How
many large T-shirts did Mary order?
b) How many medium size T-shirts
did Mary order?
11.
One
Aluminum alloy is 35% aluminum. Another aluminum alloy is 55% aluminum. How
much alloy each should be used to make 1000g of an aluminum alloy that is 40%
aluminum?
12.
Some
students at Park Collegiate held a food sale for a school project. They charge
$8 for apple pies and $7 for lemon pies.
They sold a total of 83 pies and earned $624. How many of each type of pie did
they sell?
13.
Aljohn
and Tamia are planning a trip for their class. For one option each student will
pay $875 for their meals a day and seven nights accommodation. For the second
option, each student will pay $770 for two meals in a day and seven nights’ accommodation.
a)
What
is the cost per meal?
b) What is the cost per day for accommodation?
14.
Ian
canoes 20km downstream in 5 h. On the return trip it takes him 8 h to travel 16
km. Determine his average canoeing speed and the speed of the current.
15.
With
a tailwind, a plane flew the 1800 km from Saskatoon to Toronto in 3 h. The
return flight against the wind took 4 h. Find the average speed of the plane
and the wind speed.
C.
16. Doan has some 15-karat gold (15/24) pure gold and some
10-karat gold (10/24 pure gold). What mass each type of gold should he mix to
make 500g of 14-karat gold (14/24 pure gold)?17. Bob jogged for 2 h then walked
3 h, covering of distance of 31 km. The next day he jogged for 3 h then he
walked for 2 h, covering a distance of 34 km.
Assume that his
running and jogging speeds were the same both days.
a)
Find
the speed at which Bob jogged
b) Find the speed at
which Bob walked.
18. Tera is a member of the environmental
club. Yesterday there were 220 batteries in the re-cycling bin. Tera claims that the number of AAA-size
batteries.
a) How many AA batteries were in the recycling
bin yesterday?
b) How many AAA batteries were in the
cycling bin yesterday?
19. A
flower garden has three times as many red roses as pink roses. Twice the number
of red roses is equal to four times the number of golden of pink roses
increased by ten.
a)
How many red roses are there?
b) How many pink roses are there/
20. In
the dog park today there are twice as many shelters as golden retrievers. Three
times the number of golden retrievers added to two times the number of shelties
is 21.
a) How many shelties are there?
b) How many golden retrievers are there?
***
CHAPTER
1 Review
1.
Translate
each sentence into an algebraic expression.
a)
Three
more than five times a number.
b) Five less than one third of a
value.
c)
One
number increased by four times another number.
d) A value decreased by the
fraction three quarters.
2.
Translate
each sentence into an equation. Tell how many you are assigning the variables
in each.
a)
Three
times a number increased by four, is one half the same number, decreased by
one.
b) Hannah’s age increased by five
is twice Jordan’s age, decreased by 7.
c)
Zoe
has nickels and quarters that total $1.65 in her piggy bank.
3.
Use
Technology
a)
Use
graphing calculator or The Geometric Sketchpad ® to find the point of
intersection of the lines;
y = 4x +1 and y = – (1/4) x –2, round the answer nearest 100th.
b) Use graphing calculator or The Geometric Sketchpad
® to find the point of intersection of the lines;
y = 5x +3 and y = –(1/5)
x –1, round the answer nearest 100th.
4.
Solve
each linear system by substitution, check solutions.
a)
x
+ y =5 and y = x –3
b) x –y = 7 and y = –x + 5
c)
2x
+ y =1 and x –3y = 4
d) 3x –4y = 19 and x + 2y = 3
e)
x
+ 3y = 15 and 4x –y = 8
f)
5x
–y = 4 and –x + y =2
5.
There
are 28 fish in an aquarium. There are eight more goldfish than neon tetras. How
many goldfish and neon tetras are in the aquarium?
6.
Madeline
likes to rent a digital a camera. One company charges a flat rate of $75 /day.
A second company charges $35 / day plus $5 /h.
a)
Write
two equations to represent the information.
b) Solve the linear system to
find number of hours for which the cost of renting is the same for both
companies.
7.
Which
is not an equivalent equation for 6x + 3y =15?
A
2x + y =5
B
12x + 6y = 30
C
9x + 6y = 18
D
x + (1/2) = 5/2
8.
a)
If y = 4x –5 and 5y = kx –25 are equivalent linear equations, what is k?
b)
If y = 6x + 2 and 4y = 24x + k are the equivalent linear equations, what is the
value of k?
9.
The
total number of graphing calculators and scientific calculators Charlotte’s
classroom is 14.
a)
State
how you will assign variables.
b) Write an equation to represent
this situation. Then write an equivalent linear equation.
10.
Find
the point of intersection of each pair of lines using the method of elimination,
check solution
a) x + y = y and 3x –y = 11
b) 5x + 3y = 9 and 2x –3y = 12
c) x + 3y = 11 and –x +4y = –4
d) 4x + 5y = 18 and 4x + 2 = 2
11. Solve each linear equation. Check solution
a) 5x + 4y = 20 and 4x –3y =
16
b) 5a + 3b =4 and 2a –4b = –1
c) 0.3m + 0.2n = 0.8 and 0.5m –0.3n
= 0.7
d) 2(x +3) + 4(y –2) = 10 and
3(x –2) +(x +2) =8
12. One type of fertilize has 40%
nitrogen and a secured type of fertilizer has 20% nitrogen. How much of each
type of fertilizer should be mixed to make 800kg fertilizer of 25% nitrogen.
13. Use Technology; The BLUE club Service
charges $6 plus #2.00 km travelled. GREEN’s Taxi Service charges $4 plus $2.50/
km.
a) For what distance is the
charge the same using either taxi company?
b) For what number of
kilometers would you choose the BLUE Club Service?
c) For what number of kilometers would you choose
the GREEN’s Taxi Service?
14. There are 5 books in Japan’s
library. There are more 28 more fiction books than non-fiction books.
a) How many fiction books are in Japan’s
library?
b) How many non-fiction books
are in Jean’s library?
15. Murray invests his summer earnings
of $2440. He invests part of the money at 8% years. And the rest at 7.5% /
year. After one year, these investments earn$193 in simple interest.
How much did the invest at each
rate?
116.
A
speedboat took 3 h to travel a distance of 60 km up a river, against the
current. The return trip took 2 h. Find the average speed of the boat in still
watere and the speed of the current.
****
No comments:
Post a Comment