Transcript Chapter 1 Section 3 (Solving Equations)

Motivation
• Math Magic
• Choose a number.
DRILL
Solve for x in each equation:
1) x + 13 = 20
2) x – 11 = -13
3) 4x = 32
DRILL
Solve for x in each equation:
1) x + 13 = 20
- 13 - 13
x
=7
2) x – 11 = -13
+ 11 + 11
x =-2
3) 4x = 32
Divide by 4
x=8
Solving Two-Step Equations
Algebra I
Day 25
Solving Two-Step Equations
• To solve you must still isolate the
variable by using opposite
operations.
• When you have more than one step
you must use the order of
operations in reverse order.
• Ex: (add/subtract)(multiply/divide)
(exponents)(parentheses)
Examples
1) 2x + 5 = 17
-5
-5
2x
= 12
Divide by 2 on both sides
x
= 6
DRILL
Solve for x in each equation:
1) 2x + 13 = 27
2) 3x – 7 = -13
x
3)
8  3
5
Solve for x in each equation:
1) 2x + 13 = 27
- 13 - 13
2x = 14
Divide both sides by 2
x=7
Solve for x in each equation:
2) 3x – 7 = -13
+7 +7
3x = -6
Divide both sides by 3
x = -2
x
8  3
5
x
88  38
5
x
 11
5
x
5  11(5)
5
x  55
Solving Multi-Step
Equations
Algebra I
Day 27
Combining Like-Terms
• Like terms are terms that have
the exact same exponents and
variables.
• When you add or subtract like
terms you simply add/subtract the
numbers in front of the variables
(coefficients) and keep the
variables and exponents the
same.
Example
4x + 7 + 3x – 2 = 33
7x + 5
= 33
-5
-5
7x
= 28
Divide both sides by 7
x
=4
Distributive Property
• When you have a number (term) in
parentheses next to an expression
you must multiply the number (term)
out front with each part of the
expression inside the parentheses.
• Ex:
2(3x + 4) = 6x + 8
Examples
1) 2(x + 5) = 34
2x + 10 = 34
- 10 - 10
2x
= 24
Divide by 2 on both sides
x
= 12
Lesson 1-1 Expressions and Formulas
Lesson 1-2 Properties of Real Numbers
Lesson 1-3 Solving Equations
Lesson 1-4 Solving Absolute Value Equations
Lesson 1-5 Solving Inequalities
Lesson 1-6 Solving Compound and Absolute
Value Inequalities
Example 1 Verbal to Algebraic Expression
Example 2 Algebraic to Verbal Sentence
Example 3 Identify Properties of Equality
Example 4 Solve One-Step Equations
Example 5 Solve a Multi-Step Equation
Example 6 Solve for a Variable
Example 7 Apply Properties of Equality
Example 8 Write an Equation
Write an algebraic expression to represent 3 more
than a number.
Answer:
Write an algebraic expression to represent 6 times the
cube of a number.
Answer:
Write an algebraic expression to represent the square
of a number decreased by the product of 5 and the
same number.
Answer:
Write an algebraic expression to represent twice the
difference of a number and 6.
Answer:
Write an algebraic expression to represent each
verbal expression.
a. 6 more than a number
Answer:
b. 2 less than the cube of a number
Answer:
c. 10 decreased by the product of a number and 2
Answer:
d. 3 times the difference of a number and 7
Answer:
Write a verbal sentence to represent
Answer: The sum of 14 and 9 is 23.
.
Write a verbal sentence to represent
Answer: Six is equal to –5 plus a number.
.
Write a verbal sentence to represent
Answer: Seven times a number minus 2 is 19.
.
Write a verbal sentence to represent each equation.
a.
Answer: The difference between 10 and 3 is 7.
b.
Answer: Five is equal to the sum of 2 and a number.
c.
Answer: Three times a number plus 2 equals 11.
Name the property illustrated by the statement
if xy = 28 and x = 7, then 7y = 28.
Answer: Substitution Property of Equality
Name the property illustrated by the statement
.
Answer: Reflexive Property of Equality
Name the property illustrated by each statement.
a.
Answer: Symmetric Property of Equality
b.
Answer: Transitive Property of Equality
Solve
. Check your solution.
Original equation
Add 5.48 to each side.
Simplify.
Check:
Original equation
Substitute 5.5 for s.
Simplify.
Answer: The solution is 5.5.
Solve
. Check your solution.
Original equation
Multiply each side by
multiplicative inverse of
Simplify.
the
Check:
Original equation
Substitute 36 for t.
Simplify.
Answer: The solution is 36.
Solve each equation. Check your solution.
a.
Answer: –2
b.
Answer: 15
Solve
Original equation
Distributive and
Substitution Properties
Commutative, Distributive,
and Substitution Properties
Addition and
Substitution Properties
Division and
Substitution Properties
Answer: The solution is –19.
Solve
Answer: –6
Geometry The area of a trapezoid is
where A is the area, b1 is the length of one base, b2
is the length of the other base, and h is the height of
the trapezoid. Solve the formula for h.
Area of a trapezoid
Multiply each side by 2.
Simplify.
Divide each side by
Simplify.
.
Answer:
Geometry The perimeter of a rectangle is
where P is the perimeter, is the length, and w is the
width of the rectangle. Solve the formula for w.
w
Answer:
Multiple-Choice Test Item
what is the value of
A
B
C
D
Read the Test Item
You are asked to find the value of the expression 4g – 2.
Your first thought might be to find the value of g and then
evaluate the expression using this value. However, you
are not required to find the value of g. Instead, you can
use the Subtraction Property of Equality on the given
equation to find the value of 4g – 2.
Solve the Test Item
Original equation
Subtract 7 from each side.
Answer: B
Multiple-Choice Test Item
what is the value of
A 12
B 6
C –6
D –12
Answer: D
Home Improvement Carl wants to replace the 5
windows in the 2nd-story bedrooms of his home.
His neighbor Will is a carpenter and he has agreed
to help install them for $250. If Carl has budgeted
$1000 for the total cost, what is the maximum
amount he can spend on each window?
Explore Let c represent the cost of each window.
Plan
The number
the cost
the cost for a
the total
of windows times per window plus
carpenter
cost.
equals
5
c
+
250
=
1000
Solve
Original equation
Subtract 250 from each side.
Simplify.
Divide each side by 5.
Simplify.
Answer: Carl can afford to spend $150 on each window.
Examine The total cost to replace five windows at $150
each is 5(150) or $750. Add the $250 cost of
the carpenter to that, and the total bill to
replace the windows is 750 + 250 or $1000.
Thus, the answer is correct.
Home Improvement
Kelly wants to repair
the siding on her
house. Her contractor
will charge her $300
plus $150 per square foot of siding. How much
siding can she repair for $1500?
Answer: 8 ft2