Transcript Solving equations - KTruitt

Solving Equations Using
Addition and Subtraction
Objectives:
• A.4f Apply these skills to solve practical
problems.
• A.4b Justify steps used in solving equations.
• Use a graphing calculator to check your
solutions.
To Solve an Equation means...
• To isolate the variable having a
coefficient of 1 on one side of the
equation.
• Ex: x = 5 is solved for x.
• y = 2x - 1 is solved for y.
Addition Property of Equality
For any numbers a, b, and c, if
a = b, then a + c = b + c.
What it means:
You can add any number to
BOTH sides of an equation and
the equation will still hold true.
An easy example:
• Would you ever
We all know that 7 = 7. leave the house
with only one shoe
on?
Does 7 + 4 = 7? NO!
• Would you ever put
blush on just one
But 7 + 4 = 7 + 4.
cheek?
The equation is still
• Would you ever
true if we add 4
shave just one side
of your face?
to both sides.
Let’s try another example!
x - 6 = 10
Add 6 to each
side.
x - 6 = 10
+6 +6
x = 16
• Always check your
solution!!
• The original problem
is x - 6 = 10.
• Using the solution
x=16,
Does 16 - 6 = 10?
• YES! 10 = 10 and our
solution is correct.
What if we see y + (-4) = 9?
Recall that y + (-4) = 9 • Check your
solution!
is the same as y - 4 = 9.
•
Does
13
4
=
9?
Now we can use the
• YES! 9=9 and
addition property.
our
solution
is
y-4=9
correct.
+4 +4
y = 13
How about -16 + z = 7?
• Remember to always • Check you solution!
use the sign in front
of the number.
• Does -16 + 23 = 7?
• Because 16 is
negative, we need to
add 16 to both sides. • YES! 7 = 7 and our
solution
is
correct.
• -16 + z = 7
+16
+16
z = 23
A trick question...
-n - 10 = 5
+10 +10
-n = 15
• Do we want -n? NO,
we want positive n.
• If the opposite of n
is positive 15, then n
must be negative 15.
• Solution: n = -15
• Check your
solution!
• Does -(-15)-10=5?
• Remember, two
negatives = a
positive
• 15 - 10 = 5 so our
solution is correct.
Subtraction Property of Equality
• For any numbers a, b, and c,
if a = b, then a - c = b - c.
What it means:
• You can subtract any number from
BOTH sides of an equation and the
equation will still hold true.
3 Examples:
1) x + 3 = 17
-3 -3
x = 14
• Does 14 + 3 = 17?
2) 13 + y = 20
-13
-13
y=7
• Does 13 + 7 = 20?
3) z - (-5) = -13
• Change this equation.
z + 5 = -13
-5 -5
z = -18
• Does -18 -(-5) = -13?
• -18 + 5 = -13
• -13 = -13 YES!
Try these on your own...
x + 4 = -10
x – 14 = -5
y – (-9) = 4
3 – y= 7
12 + z = 15
-5 + z = -7
The answers...
x = -14
x=9
y = -5
y = -4
z=3
z = -2
Solving Equations Using
Multiplication and Division
Objectives:
• A.4f Apply these skills to solve practical
problems.
• A.4b Justify steps used in solving equations.
• Use a graphing calculator to check your
solutions.
Remember,
To Solve an Equation means...
To isolate the variable having a coefficient
of 1 on one side of the equation.
Ex: x = 5 is solved for x.
y = 2x - 1 is solved for y.
Multiplication
Property of Equality
For any numbers a, b, and c, if a = b,
then ac = bc.
What it means:
You can multiply BOTH sides of an
equation by any number and the
equation will still hold true.
An easy example:
We all know that 3 = 3.  Would you ever put
deodorant under just one
arm?
Does 3(4) = 3? NO!
But 3(4) = 3(4).
The equation is still
true if we multiply
both sides by 4.
 Would you ever put nail
polish on just one hand?
 Would you ever wear just
one sock?
Let’s try another example!
x=4
2
Multiply each side
by 2.
(2)x = 4(2)
2
x=8
• Always check your solution!!
• The original problem is
x=4
2
• Using the solution x = 8,
Is x/2 = 4?
• YES! 4 = 4 and our solution
is correct.
What do we do with negative fractions?
Recall that
x x
x
 

5
5
5
x
3.
Solve
5
Multiply both
sides by -5.
• The two negatives will
cancel each other out.
• The two fives will
cancel
 xeach other out.
 3(-5)
(-5)
5
• x = -15
• Does -(-15)/5 = 3?
Division Property of Equality
 For any numbers a, b, and c (c ≠ 0),
if a = b, then a/c = b/c
What it means:
 You can divide BOTH sides of an
equation by any number - except zeroand the equation will still hold true.
 Why did we add c ≠ 0?
2 Examples:
1) 4x = 24
Divide both sides by 4.
4x = 24
4
4
x=6
2) -6x = 18
Divide both sides by -6.
-6y = 18
-6
-6
y = -3
• Does 4(6) = 24? YES! • Does -6(-3) = 18? YES!
A fraction times a variable:
The two step method:
Ex: 2x = 4
3
1. Multiply by 3.
(3)2x = 4(3)
3
2x = 12
2. Divide by 2.
2x = 12
2
2
x=6
The one step method:
Ex: 2x = 4
3
1. Multiply by the
RECIPROCAL.
(3)2x = 4(3)
(2) 3
(2)
x=6
Try these on your own...
x=3
7
4w = 16
y=8
-2
2x = 12
3
-2z = -12
3x = 9
-4
The answers...
x = 21
w= 4
y = -16
x = 18
z=6
x = -12
Solving Multi-Step Equations
Objectives:
• to solve equations involving more than one operation.
• to solve problems by working backward.
• A.4d Solving multistep linear equations.
• A.4f Solve real-world problems involving
equations.
• A.4b Justify steps used in solving equations.
To Solve: Undo the operations by
working backward.
Ex: x + 9 = 6
5
Ask yourself:
• What is the first thing
we are doing to x?
• The second thing?
Recall the order of
operations as you
answer these questions.
• dividing by 5
• adding 9
To undo these steps, do
the opposite operations
in opposite order.
The DO-UNDO chart
Use a chart as a shortcut to
answering the questions.
DO UNDO
• ÷5
-9
• +9
·5
Follow the steps in the
‘undo’ column to isolate
the variable.
Ex: x + 9 = 6
5
• First subtract 9.
x+9-9=6-9
5
x = -3
5
• Then multiply by 5.
(5) x = -3(5)
5
x = -15
Let’s try another!
Complete the do-undo chart.
DO UNDO
• -2
·3
• ÷3
+2
To solve for d:
• First multiply by 3.
• Then add 2.
Ex: d - 2 = 7
3
(3) d - 2 = 7(3)
3
d - 2 = 21
d - 2 = 21
+2 +2
d = 23
Here’s a tricky one!
Remember to always use the
sign in front of the number.
DO UNDO
• ÷ -7
-3
• +3
· -7
To solve for a:
• First subtract 3.
• Then multiply by -7.
Ex: 3 - a = -2
7
•
3 - a = -2
7
-3
-3
- a = -5
7
• (-7)(- a) = (-5)(-7)
7
a = 35
Try a few on your own.
• 5z + 16 = 51
• 14n - 8 = 34
• 4b + 8 = 10
-2
The answers:
DO
• ·5
• +16
• z=7
UNDO
- 16
÷5
DO
UNDO
• · 14 +8
• -8
÷ 14
• n=3
DO
• ·4
UNDO
· -2
• +8
• ÷ -2
-8
÷4
• b = -7
Solving Equations with the
Variable on Both Sides
Objectives:
• to solve equations with the variable on both sides.
• to solve equations containing grouping symbols.
To solve these equations,
•Use the addition or subtraction
property to move all variables to one
side of the equal sign.
Let’s see a few examples:
1) 6x - 3 = 2x + 13
Be sure to check your
answer!
Let’s try another!
2) 3n + 1 = 7n - 5
Check:
Here’s a tricky one!
3) 5 + 2(y + 4) = 5(y - 3) + 10
Check:
\
Let’s try one with fractions!
4) 3
1
1
3
 x  x
8 4
2
4
Steps:
•Multiply each term
by the least common
3
1
1
3 denominator (8) to
(8)  (8) x  (8) x  (8)
8
4
2
4 eliminate fractions.
3 - 2x = 4x - 6
3 = 6x - 6
9 = 6x so x = 3/2
•Solve for x.
•Add 2x.
•Add 6.
•Divide by 6.
Two special cases:
6(4 + y) - 3 = 4(y - 3) + 2y
3(a + 1) - 5 = 3a - 2
24 + 6y - 3 = 4y - 12 + 2y
3a + 3 - 5 = 3a - 2
21 + 6y = 6y - 12
- 6y - 6y
21 = -12 Never true!
21 ≠ -12 NO SOLUTION!
3a - 2 = 3a - 2
-3a
-3a
-2 = -2 Always true!
We write IDENTITY.
Try a few on your own:
• 9x + 7 = 3x - 5
• 8 - 2(y + 1) = -3y + 1
• 8-1z=1z-7
2
4
The answers:
• x = -2
• y = -5
• z = 20
A number is doubled and then increased by
seven. The result is ninety-three.
What is the original number?
Six less than five times a number is
the same as seven times the number.
What is the number?
Brad is a waiter, and he gets paid $5.75 per hour,
and he can keep his tips. He knows his tips average
$8.80 per table.
If he worked an eight-hour shift and took home
$169.20, how many tables did he serve?
In warmer climates, approximate temperature predictions can be made by
counting the number of chirps a cricket makes during a minute. The
temperature (in Fahrenheit) decreased by 40 is equivalent to one-forth of
the number of cricket chirps in a minute.
(a) Write an equation for this relationship.
(b) Approximately how many chirps per minute should be recorded if the
temperature is 90 F?
(c) If a person recorded 48 cricket chirps in a minute,
what would the temperature be?
Two trains leave a train station at the same time.
One train travels east at 50 mph. The other train
travels west at 55mph.
In how many hours will the two trains be 315
miles apart?