CHAPTER 2 STUDY GUIDE
Unit Essential Question: How does solving linear equations help solve
real-life problems?
1-7: Essential Question: How can the distributive property be used to simplify
expressions?
How do you combine like terms?
2-1/2: Essential Question: How do you use inverse operations to solve one-step
equations?
2-3/4: Essential Question: How do you use the order of operations to solve multi- step
equations?
How do you solve equations that contain variables terms on
both sides of the equal sign?
2-6: Essential Question: How do you write and solve proportions for real-life
problems?
STANDARDS
A1.1.2.1.1 Write, solve and/or apply a linear equation (including problem situations).
A1.1.2.1.2 Use and/or identify an algebraic property to justify any step in an equationsolving process (linear equations only).
1
Are there parentheses?
No
SOLVING LINEAR EQUATIONS USING A FLOWCHART
Yes
Distribute
Can you combine like terms on either side of the equation? (Variables must be EXACTLY the same)
No
Yes
Combine like terms
Are there variables on both sides of the equation?
No
Yes
Move them to one side of the equation using the (+/-) inverse
operations. Combine/simplify the variable side of the equation.
Is there a number (constant) NOT attached to the variable on the same side as the variable?
No
Yes
Move the number to the OTHER side of the equation using the (+/-) inverse
operations. Combine/simplify the number (constant) side of the equation.
Is there a number attached to the variable?
No
Problem is solved. Check your work.
Yes
1. If attached by multiplication, then divide.
2. If attached by division, then multiply.
3. If attached by a fraction, then multiply by its reciprocal.
2
1-7: Properties of Real Numbers:
Commutative Property of addition: a + b = b + a
Commutative Property of multiplication: ab = ba
Associative Property of addition: (a + b) + c = a + (b + c)
Associative Property of multiplication: (ab)c = a(bc)
Identity Property of addition: a + 0 = a
Identity Property of multiplication: a • 1 = a
Inverse Property of addition: a + (-a) = 0
1
Inverse Property of multiplication: a( ) = 1
a
Distribution Property: a(b + c) = ab + ac
Examples:
1. Which property of real numbers is illustrated by the equation - 3 + 3 = 0?
a. additive identity
b. commutative property for addition
c. associative property of addition
d. additive inverse
2. Which equation illustrates the associative property of addition?
a. x + y = y + x
b. 3(x + 2) = 3x + 6
c. (3 + x) + y = 3 + (x + y)
d. 3 + x = 0
3. Which expression is an example of the commutative property?
a. xy = yx
b. (x + y) + z = x + (y + z)
c. x • 1 = x
d. x(y + z) = xy + xz
Distributive Property:
4. 3(x + 2) = _________________
5. -2(4 – y) = _____________________ = _______________________
6. -5(x2 + 3x – 2) = ____________________________ = _________________________
7. (b – 3)6 = ______________ = ___________________ = _____________________
3
Combining like terms:
Definition: Terms with same variable(s) raised to the same exponent.
8. 3x − 7x = ________________ = _________________
9. 4x2 + 8x2 – 6x2 = ______________________ = _______________________
10. 9m3 + 4m2 = ____________________________ = __________________
11. 4x + 5y − 2x + y = _________________________
12. 2a + 3b – 6ab + 5b – 3a + 6ab = ___________________
Combining like terms with distribution:
13. 7x – 2(x + 4) = _________________ = ________________________
14. 3(x -2) – 8x = __________________ = ________________________
4
2:1-2 Solving one-step equations
Equation: Mathematical statement that two expressions are equal.
Solution: A value of a variable that makes the equation true.
Inverse Operations:
Solving equations using addition or subtraction:
Ex 1: x + 7 = 10
Ex 3: y −
1
2
=
3
3
Ex 2:
3=a−8
Ex 4:
-4 + y = 12
Ex 8:
5x = -25
Solving equations by multiplying and dividing
Ex 7:
Ex 9:
t
= -6
3
1
x=2
3
Ex 10:
5
4
2
= - y
5
7
2-3: Solving 2-step and multi-step equations
Isolate the variable by doing the order of operations backwards:
Subtraction/Addition then Division/Multiplication then Exponents then Parentheses
Ex 1:
-4 + 7x = 3
Ex 2:
Ex 3:
n
+2=2
7
Ex 4:
1
2
3
x + 4 = -5
5x – 3 – 2x = 9
Ex 6: 7(2x – 3) + 3 = 24
Ex 5: -2(b + 5) = -6
Ex 7: Tom is thinking of a number such that 6 less than the product of 7 and this number
is 85. Which number is he thinking of?
a. 11 2
7
b. 84
c. 13
d. 637
.Ex 8: Which equation can be used to find the value of x?
a. 2x = x + 15
c. 2x + x + 15 = 180
x
x
b. 2x + x + 15 = 90
d. 2x(x + 15) = 0
x
2x
6
15
2-6: Proportions:
Definitions:
Proportion: A statement that two ratios are equivalent.
a
c
Cross Product: In a proportion
= , the products a • d = b • c.
b
d
3
5
3
4
Ex 1:
=
Ex 2:
=
9
m
9
y
Ex 3:
2
6
=
y 3 7
Ex 4:
8
3
=
x  10
 12
Ex 5: A student can read 7 pages in 10 minutes. How many pages can the student
read in 40 minutes?
Ex 6: A recipe call for 2 cups of flour to make 12 biscuits. How much flour is
needed to make 30 biscuits?
Ex 7: The following shapes are similar figures. Write and solve a proportion to
find the value of x.
19
6
40 x
12
7
2-4: Solving Equations with Variables on Both Sides
***Use the flow chart on page 2 to solve these multi-step equations.
Ex 1:
2x – 7 = 3x + 4
Ex 2: 5y = 2y - 42 – 3y
Ex 3:
2a + 3 −8a = 8 – a
Ex 4: -2(3 − d) = 4 + 3d
Ex 5:
2
1
x–3=3x
5
Ex 6: 4y – 3(2 – y) = 1y
8