Download Equations

UNIT 6. EQUATIONS
1. EQUATION AND EXPRESSION
In the lesson on Algebraic Representations we looked at writing mathematical
expressions.
We are ready now to write mathematical equations.
Expressions:
10, 2 + y, z,
7xy,
5x - 6, 2(a +
3),
x2 + 4x + 4
Equations:
"Expressions" do NOT have an
equal sign.
"Equations" have an
EQUAL sign.
2 + y = 10
5x - 6 = 41
x2 + 4x + 4
=0
"Expressions" and "equations" refer to two different entities in mathematics.
If you are dealing with an EQUAL SIGN, you are dealing with an "equation".
EQUAtion ↔ EQUAL sign
An equation is a sentence where two algebraic expressions are equal.
1
Examples
Translate the following problems into equations:
HINT: If you are having trouble determining what the equation might be, make
a numerical problem to see how the numbers in the problem are related to one
another.
1.
Cost for cable TV service in a certain city
is 45€ for the installation and 24€ a month
for service.
Let c = cost, m = number of months
c = 24m + 45
Number problem: for 3 months the cost would
be: Cost = 24 • 3 € + 45€. Replace 3 with m.
2.
If golf balls cost 1.25€ and putters cost 79€,
how many of each can the golf team
purchase for 150€?
Number problem: if I purchase 2 golf balls and
3 putters the cost will be: Cost = 1.25•2 + 79•3
Replace 2 with g (number of golf balls), 3 with
p (number of putters), and Cost with 150€.
3.
A cookie recipe calls for twice as many
chocolate chips as walnuts.
Number problem: if a recipe calls for 10
walnuts, it calls for 20 chocolate chips. The
number of chocolate chips is twice the number of
walnuts.
5.
1.25g + 79p = 150
H2O means that water contains two
hydrogen atoms and one oxygen atom.
Number problem: In water, if there is one
oxygen atom, there are 2 hydrogen atoms; if
there are two oxygen atoms, there are 4
hydrogen atoms and so on. The number of
hydrogen atoms is twice the number of oxygen
atoms.
4.
Let g = number of golf ball,
p = number of putters
What is the total weight of a filled hot tub,
if the tub weighs 175 kg pounds and the
water weighs 5 kg per liter?
Number problem: if it takes 100 liters to fill
the hot tub, the total weight will be:
Total = 5 • 100 + 175 Replace 100 with g, for
the number of liters.
Let O = number of oxygen atoms,
H = number of hydrogen atoms
H = 2O
Let c = number of chips,
w = number of walnuts
c = 2w
Let t = total weight,
g = number of liters
t = 5g + 175
2
Translate each of the following into an equation and solve if asked.
1.
2.
3.
Using the variable x to represent a number,
write an algebraic equation for:
"twice a number increased by four equals 18".
Answer
Using the variable x to represent a number,
write an inequality for:
"three times a number decreased by four is five time a
number".
A formula about triangles defines a quantity s to be equal to
one-half the sum of the three sides (a, b and c) of a triangle.
Write this formula in mathematical notation.
Solve the formula for a when s = 6, b = 4 and c = 5.
Answer
Answer
Answer
4.
A student receives $10 a week for lunch in school. The cafeteria
charges $1.60 for a plate lunch. Write an equation to help the
student calculate the amount of money, m, left after d days,
assuming the student purchases the plate lunch each day.
5.
A recipe calls for 1/2 cup of chocolate chips. Write a equation Answer
to determine the number of cups, c, of chocolate chips needed
to make the recipe n times.
Find the number of cups of chips needed to double the
recipe.
http://www.regentsprep.org/Regents/math/ALGEBRA/AE1/PAlgTrans.htm
3
2. PARTS OF AN EQUATION
What is an Equation
An equation says that two things are equal. It will have an equals sign "=" like this:
x
+
2
=
6
That equations says: what is on the left (x + 2) is equal to what is on the right (6)
So an equation is like a statement "this equals that"
Parts of an Equation
So that people can discuss equations, there are names for different parts (better than saying
"that thingy there"!)
Here we have an equation that says 4x-7 equals 5, and all its parts:
A Variable is a symbol for a number we
don't know yet. It is usually a letter like x or
y.
A number on its own is called a Constant.
A Coefficient is a number used to multiply
a variable (4x means 4 times x, so 4 is a
coefficient)
An Operator is a symbol (such as +, ×, etc)
that represents an operation (ie you want to
do something with the values).
A Term is either a single number or a
variable, or numbers and variables
multiplied together.
An Expression is a group of terms (the
terms are separated by + or - signs)
So, now we can say the equation has two sides. First side before the equal sign and second
side after the equal side.
4
3. SOLVING LINEAR EQUATIONS
Solving linear equations is just a matter of undoing operations that are being done to the
variable. The task is always to isolate the variable -- get the variable ALONE on one side
of the equal sign.
Then, to solve an equation, we have to find the correct value for the letter
contained in the equation. To do this, rearrange the equation so that only the
letter is on one side of the equal sign. Remember, when we move a number to
the other side of the equal sign, we must also change the number's sign.
A Puzzle
What is the missing number?
-
2
=
4
OK, the answer is 6, right? Because 6-2=4. Easy stuff.
Well, in Algebra we don't use blank boxes, we use a letter (usually an x or y, but
any letter is fine). So we would write:
x
-
2
=
4
It is really that simple. The letter (in this case an x) just means "we don't know
this yet", and is often called the unknown or the variable.
And when you solve it you write:
x
=
6
Why Use a Letter?
Because:
it is easier to write "x" than drawing empty boxes (and
easier to say "x" than "the empty box")
if there were several empty boxes (several "unknowns") we
can use a different letter for each one.
5
How to Solve
Algebra is just like a puzzle where you start with something like "x-2 = 4" and
you want to end up with something like "x = 6".
But instead of saying "obviously x=6", use this neat step-by-step approach:


Work out what to remove to get "x = ..."
Remove it by doing the opposite (adding is the opposite of subtracting)
Example 1: In a simple equation, you may only have to
undo one operation to solve the equation.
Solve this equation for x:
x+3=8
x+3=8
x =8-3
The variable is x and we need to get it alone.
x =5
Check your answer:
You will always know if your answer is correct by doing a
simple "check" -- substitute your answer into the original
equation and see if the result is true.
Check:
x+3=8
5+3=8
8 = 8 true
1. Solve for x and check: x + 5 = 12
2. Solve the following:
a) x + 9 = 11
b) x - 10 = 0
e) 9 + x = 16
f) x + 5 = 13
i) x - 9 = -7
j) -10 + x = 0
c) x + 7 = 17
g) -9 + x = -2
d) 2 + x = 8
h) x + 7 = 9
6
Another Puzzle
What is the missing number?
4
·
=
8
The answer is 2, right? Because 2 ·4 = 8.
Well, in Algebra we don't use blank boxes, we use a letter. So we might write:
x
4
·
=
8
But remember in Algebra we don't use the multiply symbol (·) between numbers
and letters, we simply put the number next to the letter to mean multiply:
4x
=
8
You would say in English "four x equals eight", meaning that 4 ·'s make 8. And
the answer would be written:
x
=
=
2
How to Solve
On the previous page we showed this neat step-by-step approach:


Work out what to remove to get "x = ..."
Remove it by doing the opposite +
Example 2: In an equation which has more than one
operation, we have to undo the operations in the correct
order. First, undo addition or subtraction, then undo
multiplication or division.
7
Solve this equation for x:
5x - 2 = 13
The variable is x.
The question is leaving the x term alone in one side,
and the constants in the other, remember moving any
term across the "equal sign bridge changes the term's
sign (like paying a toll).
5x - 2 = 13
5x = 13 + 2
5x = 15
Then operate the constants.
Finally, undo the multiplication by dividing by the
coefficient of the x term.
x =
=3
Check:
5x - 2 = 13
5(3) - 2 = 13
15 - 2 = 13
13 = 13 true
Check your answer:
3. Solve for x and check: 3x + 5 = 11
4. Solve the following:
a) 10x + 7 = 37
b) 7 + 5x = 37
c) 3x - 10 = 20
d) -3 + 7x = 18
e) 9x - 8 = 55
f) 4x + 32 = 40
g) 8x - 3 = 69
h) 36 + 9x = 117
i) 7x + 9 = 65
j) 9x - 27 = 54
k) 3x + 2 = 17
l) 4x + 5 = 29
http://www.bbc.co.uk/schools/ks3bitesize/maths/algebra/equations1/activity.shtml
TEST
http://www.bbc.co.uk/apps/ifl/schools/ks3bitesize/maths/quizengine?quiz=equations1&templateStyle=maths
8
5. Write each sentence as an algebraic equation and solve.
Three times a number is twenty-one.
A number increased by seventeen is twenty-two.
Twice a number is seventy-six.
Twelve less than a number is thirty.
Six is fifteen less than three times a number
Eighty-one is three times a number
Example 3: Suppose there are variables on both sides of the
equation. The trick now, is to get the variables on the same
side by adding them or subtracting them.
Solve this equation for x: 4x + 5 = x - 4
This question has two terms with the variable; 4x and x. We need to get the
variables combined into one term.
The question is leaving the x terms alone in one side,
and the constants in the other, remember moving any
term across the "equal sign bridge changes the term's
sign (like paying a toll).
Then operate. The sign in front of the x is implied
to be +.
4x + 5 = x - 4
4x - x = -4 -5
3x = -9
x=
Now we proceed as before.
Check your answer:
x = -3
Check:
4x + 5 = x - 4
4(-3) + 5 = -3 -4
-12 + 5 = -7
-7 = -7 true
6. Solve for y: a) 2y + 1.8 = 4y - 4.4
b) 7y + 5 - 3y + 1 = 2y + 2
http://kent.skoool.co.uk/viewdetails_ks3.aspx?id=554
9
7. Solve:
a) 30 + 4x = -5x -4
b) 34 - x = x + 10
c) 300 - 120a = -60a
d) 80 = 50 -10x + x
e) -14b = -2b -3b
f) -13x + 2 = 3x - 7
g) 3x + 8x - 2 = 7x + 3
h) 2x + 5 = 70x - 30
Example 4: There are also equations with parentheses. The
first step in these problems is to multiply and get rid of the
parentheses.
Solve this equation for n:
3(n - 1.8) = 2n + 1
First distribute the 3 -- multiply through the
parentheses by 3.
Now proceed as normal.
3(n - 1.8) = 2n + 1
3n - 5.4 = 2n + 1
3n - 2n = 1 + 5.4
n
Check your answer:
= 6.4
Check:
3(n - 1.8) = 2n + 1
3(6.4 - 1.8) = 2(6.4) + 1
3(4.6) = 12.8 + 1
13.8 = 13.8 true
8. Solve for m and check: 8(m + 5) = 16
9. Solve the following equations:
a)
b)
c)
d)
e)
f)
g)
3( x  1)  4( x  3)  22
7( x  2)  9( x  6)  116
5( x  8)  3( x  1)  99
6(2 x  9)  2(4 x  1)  112
2(7 x  1)  8(3x  5)  304
3(2 x  9)  7( x  2)  26
8(5 x  3)  3(2 x  6)  86
h) x  3  ( x  3)  ( x  3)
i) 4  (6  x)  3 x  5  2 x  (4 x  3)
j) 1  (4  x)  3 x  1
k) 2(3x  5)  7  4 x  5( x  2)  4
l) 7( x  5)  2(4  x)  4 x  13
m) 2  ( x  1)  3x  2  4 x  3( x  1)
n) 8  5 x  3(2 x  4)
10
Example 5: The last type of equation contains fractions.
Solve for x: This problem could also be solved by multiplying each term
by the common denominator, 3, thus creating:
6x + 1 = 6
6x = 5
x = 5/6
10. Solve for x:
1.
3.
5.
7.
9.
11.
13.
+1=9
+1=8
+ 5 = 10
+7=9
- 5 = 13
- 3 = 11
-2=4
2.
4.
6.
8.
10.
12.
14.
+6=7
+ 8 = 14
+3=9
+3=7
-4=8
-1=7
- 2 = 10
11
11.
Solve the following equations:
x 1 x  2

5
7
x 1 x  2

b)
3
2
a)
x  3 x 1

4
3
2x 1 x  6

d)
3
2
c)
x  5 2x 1

6
2
3x  1 5 x  4

f)
2
7
e)
12. Solve the following equations:
a)
b)
c)
d)
e)
x  2 x 1

4
2
3
x 1 x  5

8
3
2
x 5 x 3

 2
2
6
3x  1 x  1

4
5
3
2 x  3 3x  1

7
9
12
13. Solve the following equations:
x  3 x 1 3


a)
7
2
14
2( x  1) 6  2 x

1
b)
9
3
2x
1
1  x 
c)
3
3
x  3 1  2x

d) 2( x  1) 
2
6
x  2 3( x  6)

 x  1
e)
8
4
14. Solve the following equations:
3x 1
 2
a)
f)
2 3
x 3x

1
b)
g)
3 2
x 1 x  2

2
c)
h)
2
3
6x x 1

d)
i)
5
2
2x 5
 2
4 3
x2 x
  3x
3
2
2 x  2 3x  1

2
7
2
5x  2 2 x  3

5
4
12
Worksheet
- Solving Equations
http://www.regentsprep.org/Regents/math/ALGEBRA/AE2/Worksheet%20Bingo.pdf
Answers:
1. x= 36
2. n = 11
3. n = 13
4. y = 23
5. x = 3
6. x = -3
7. y = 52
8. w = 90
9. t = 3/2
10. a = 7/3
11. y = 21
12. c = -1
13. x = 1
14. y = 10
15. y = 9
16. x = -7
17. x = 8
18. n = 16
19. y = 12
20. y = 29
21. p = 18
22. n = -8.5
23. x = 4
24. a = -26
25. c = -75
Linear equations
- Revision
http://www.slideshare.net/keithpeter/linear-equations-slide-share-version-exploded1
"I'm Thinking of a Number"
Write down the equations, solve and answer
I'm thinking of a number. If you multiply my
number by 5 and then add 6, you will get 51.
What is my number?"
"I'm thinking of a number. If you multiply my
number by 6 and then add 13, you will get
85. What is my number?"
"I'm thinking of a number. If you
multiply my number by 3 and then
subtract 5, you will get 40. What is
my number?"
"I'm thinking of a number. If you
divide my number by 2 and then
add 10, you will get 52. What is
my number?"
"I'm thinking of a number. If you divide my
number by 5 and then subtract 7, you will get
5. What is my number?"
"I'm thinking of a number. If you
multiply my number by 2 and then
add 3 times my number, you will
get 55. What is my number?"
"I'm thinking of a number. If you divide my
number by 3 and then add 4 times my
number, you will get 390. What is my
number?
"I'm thinking of a number. If you
multiply my number by 5 and then
subtract 48, you will get my
number. What is my number?"
13
Exercises - Word problems
There are two steps to solving math word problems:
1. Translate the wording into a numeric equation that combines
smaller "expressions"
2. Solve the equation!
1. If twice a number is equal to 68, find the number.
2. If a number is decreased by 4, the result is 29. Find the number.
3. If a number is increased by 8, the result is 25. Find the number.
4. If a number is divided by 9, the result is 12. Find the number.
5. If three times a number decreased by 5 equals 82, find the number.
6. If one-half of a certain number is added to one-third of the same number, the result is
10. Find the number.
7. The sum of twice a number and 13 is 75. Find the number.
8. The sum of 6 and three times a number equals 33. Find the number.
9. The difference between 55 and four times a number is equal to fifteen. Find the number.
10. The sum of 3 and six times a number is equal to 39. Find the number.
11. A rectangular paddock is twice as long as it is wide. If it has a perimeter of 570 m, find
its dimensions.
12. If the sum of two consecutive numbers is 159, find the numbers.
13. The sum of two number is 24. Their difference is 6. Find the number
14. The sum of two numbers is 66. The larger number is three less than twice the smaller
number. Find the numbers.
15. One number is four times another. If the difference between the larger and the smaller
is 15, find both numbers.
14
16. A supermarket had a one day sale on Pepperidge Farm cookies. It sold three times as
many packages of chocolate-chip cookies as oatmeal, and twice as many fudge-almond as
oatmeal. If the supermarket sold 600 packages in one day, how many packages of each did
it sell?
17. Peter's father is 45. He is 15 years older than twice Peter's age. How old is Peter?
18. A man is 27 years older than his son and 10 years from now, he will be twice as old as
his son. How old is each now?"
19. The sum of 5 and a number divided by 4 is 9. Find the number.
4. QUADRATIC EQUATIONS
4.1 Definition for a Quadratic Equation
A Quadratic Equation is an equation with an 'x²' in it and no higher power of 'x'. Obviously
it must have an equals sign also.
4.2 A quadratic equation in STANDARD FORM
(a, b, and c can have any value, except that a can't be 0.)


The letters a, b and c are coefficients (you know those values)
The letter "x" is the variable or unknown (you don't know it yet)
Here is an example:
15
More Examples of Quadratic Equations:
In this one a=2, b=5 and c=3
This one is a little more tricky:



Where is a? In fact a=1, because we don't usually write
"1x2"
b=-3
And where is c? Well, c=0, so is not shown.
Oops! This one is not a quadratic equation, because it is missing
x2 (in other words a=0, and that means it can't be quadratic)
1. Write these equations using the standard form and write the coefficients:
EQUATION
x + 2x – 1 = 0
3x2 - 2x = 5
x2 = 25
7x2 = 0
-3x(x+2) = 0
(x-1)(x+4) =1
3x2 – 5x = 0
3x(x-2) = 4
-x2 -5x + 3 = 0
STANDARD FORM
2
a=
a=
a=
a=
a=
a=
a=
a=
a=
COEFFICIENTS
b=
c=
b=
c=
b=
c=
b=
c=
b=
c=
b=
c=
b=
c=
b=
c=
b=
c=
http://kent.skoool.co.uk/content/keystage4/maths/pc/modules/solv_quad_equations/index.html
 Introduction
 Solving quadratic equations
 Solutions of quadratic equations
2. Write a quadratic equation with the following coefficients:
a) a = 4
b) a = 6
c) a = -1
d) a = -1
b = -3
b= 0
b=2
b=0
c = -2
c = -3
c=0
c=0
16
4.3 Solving quadratic equations
Type 1.
ax2 + bx = 0  c = 0
Type 2.
ax2 + c = 0  b = 0
Type 3.
TYPE 1.
ax2 + bx + c = 0
ax2 + bx = 0  c = 0
Extract common factor.
Set each factor equal to 0 and solve
for x.
x(ax +b)=0
x=0
ax + b = 0
x = b
a
List all values of x.
TYPE 2.
x = {0,
b
}
a
ax 2 + c = 0  b = 0
Isolate x2
Take the square root for solving x
List all values of x.
3. Solve the following quadratic equations:
a) y2 - 81= 0
b) a2 = 36
c) m2 = 7m
d) 5 x 2  80  0
e) 3 x 2  12  0
f) 4 x 2  16  0
g) 3x 2  147  0
h) x 2  169  0
i) 7 x 2  343
x2 =
x=
x = {+
j)
k)
l)
m)
n)
o)
c
a

c
a
c
,a
c
a
}
3 x 2  243
x 2  24  120
x 2  11x  0
 10 x 2  24 x  0
3 x 2  12 x  0
18 x  6 x 2
17
TYPE 3.
ax 2 +bx + c = 0
Now we simply replace the letters in the formula below by the correct numbers:
I don't know of an easy way to remember the Quadratic Formula, I just say to myself:
"minus b plus or minus the square root of b-squared minus four ac, all
over two a"


The ± means you need to do a plus AND a minus, and so there are normally TWO
solutions. The answers it gives are the "solutions" to the Quadratic Equation, and
are often called "roots".
Now it's time to see how the formula is used. Remember that all terms must be on
the same side of the equals sign. The strange looking symbol ± will cause there to
be two distinct solutions. We take the + sign first and the - sign second.
18
http://kent.skoool.co.uk/content/keystage4/maths/pc/modules/solv_quad_equations/index.html
 Example 1 – Question 4
4. Solve the following quadratic equations:
a) 5x² + 6x + 1 = 0
b)
c)
d)
e)
f)
g)
h)
i)
j)
k)
l)
m)
Coefficients are:
x  5x  6  0
x 2  5x  4  0
x2  x  6  0
x 2  9 x  20  0
x 2  6x  9  0
x 2  12 x  36  0
x 2  2x  5  0
x2 - 3x - 10 = 0
2y2 + 4 = 9y
2 x 2  3x  2  0
3x 2  5 x  3  0
2x 2  x  6  0
2
a = 5, b = 6, c = 1
n) 3x 2  6 x  45  0
o) 6 x 2  18 x  24  0
p)
q)
r) 4 x 2  3 x
2
s) 4 x  1  7 x 2  8 x  5
x
x
t)
x4
5. Classify depending on the number of solutions
a)
b)
c)
d)
e)
f)
g)
h)
x 2  4x  4  0
3x 2  2 x  5  0
x2  x 1  0
x 2  20 x  100  0
 3x 2  5 x  10  0
x 2  7 x  11  0
4x 2  2x  2  0
x2  x  2  0
19
WORD PROBLEMS
http://kent.skoool.co.uk/content/keystage4/maths/pc/modules/solv_quad_equations/index.html
 Real problem 1 – Exercise 3
1. The square of a number exceeds 5 times the number by 24. Find the number(s).
HINT: Translate the problem into a mathematical equation. x2 = 5x + 24
Get all terms on the same side.
2. In football, the height of the football reached during a pass can be modeled by the
equation h = -16t 2 + 28t + 6, where the height, h, is in feet and the time, t, is in seconds.
How long does it take for this ball to reach a height of 12 feet?
HINT: Substitute 12 into the equation for h.
3. The square of a number is decreased by 15. This value is twice the original number.
Find the number(s).
4. The product of two consecutive integers is 56. Find the integers.
5. During practice, a softball pitcher throws a ball whose height can be modeled by the
equation h = -16t 2 + 24t +1, where h = height in feet and t = time in seconds. How long
does it take for the ball to reach a height of 6 feet?
6. One leg of a right triangle exceeds the other leg by four cm. The hypotenuse is
20cm. Find the length of the shorter leg of the right triangle.
20