Transcript 2-1 Solving One-Step Equations

2-1 Solving One-Step Equations
Preview
Warm Up
California Standards
Lesson Presentation
2-1 Solving One-Step Equations
Warm Up
Evaluate.
2
1. – 2 + 4 1 3
3
3
3
3. 0.96 ÷ 6 0.16
5.
2. –0.51 + (–0.29) –0.8
4. (–9)(–9) 81
1
Evaluate each expression for a = 3 and b = –2.
6. a + 5 8
7. 12b –24
2-1 Solving One-Step Equations
California
Standards
Preparation for
5.0
Students solve multistep problems, including
word problems, involving linear equations
and linear inequalities in one variable, and
provide justification for each step.
Also covered:
2.0
2-1 Solving One-Step Equations
Vocabulary
equation
solution of an equation
solution set
2-1 Solving One-Step Equations
An equation is a mathematical statement that
two expressions are equal.
A solution of an equation is a value of the
variable that makes the equation true. A
solution set is the set of all solutions. Finding
the solutions of an equation is also called solving
the equation.
2-1 Solving One-Step Equations
To find solutions, perform inverse operations
until you have isolated the variable. A variable
is isolated when it appears by itself on one side
of an equation, and not at all on the other side.
Inverse Operations
Add x.
Multiply by x.
Subtract x.
Divide by x.
An equation is like a balanced scale. To keep the
balance, you must perform the same inverse
operation on both sides.
2-1 Solving One-Step Equations
2-1 Solving One-Step Equations
Writing Math
Solution sets are written in set notation
using braces, { }. Solutions may be given
in set notation, or they may be given in the
form x = 14.
2-1 Solving One-Step Equations
Additional Example 1A: Solving Equations by Using
Addition or Subtraction
Solve the equation.
y – 8 = 24
+8 +8
y = 32
Since 8 is subtracted from
y, add 8 to both sides to
undo the subtraction.
The solution set is {32}.
Check
y – 8 = 24
32 – 8
24
24
24
To check your solution,
substitute 32 for y in the
original equation.
2-1 Solving One-Step Equations
Additional Example 1B: Solving Equations by Using
Addition
Solve the equation.
4.2 = t + 1.8
–1.8
–1.8
2.4 = t
Check 4.2 = t + 1.8
4.2 2.4 + 1.8
4.2 4.2 
Since 1.8 is added to t,
subtract 1.8 from both sides
to undo the addition.
The solution set is {2.4}.
To check your solution,
substitute 2.4 for t in the
original equation.
2-1 Solving One-Step Equations
Check It Out! Example 1a
Solve the equation. Check your answer.
n – 3.2 = 5.6
+ 3.2 + 3.2
Since 3.2 is subtracted from
n, add 3.2 to both sides to
undo the subtraction.
n = 8.8
Check
The solution set is {8.8}.
n – 3.2 = 5.6
8.8 – 3.2
5.6
5.6
5.6 
To check your solution,
substitute 8.8 for n in the
original equation.
2-1 Solving One-Step Equations
Check It Out! Example 1b
Solve the equation. Check your answer.
–6 = k – 6
+6
+6
0=k
Check
Since 6 is subtracted from
k, add 6 to both sides
to undo the subtraction.
The solution set is {0}.
–6 = k – 6
–6 0 – 6
–6 –6 
To check your solution,
substitute 0 for k in the
original equation.
2-1 Solving One-Step Equations
Check It Out! Example 1c
Solve the equation. Check your answer.
6 + t = 14
–6
–6
t=
8
Check 6 + t = 14
6 + 8 14
14 14 
Since 6 is added to t, subtract 6
from both sides to undo the
addition.
The solution set is {8}.
To check your solution,
substitute 8 for t in the
original equation.
2-1 Solving One-Step Equations
2-1 Solving One-Step Equations
Additional Example 2A: Solving Equations by Using
Multiplication or Division
Solve the equation. Check your answer.
Since j is divided by 3, multiply
from both sides by 3 to undo
the division.
The solution set is {–24}.
–24 = j
Check
To check your solution, substitute
–24 for j in the original equation.
–8
–8 
2-1 Solving One-Step Equations
Additional Example 2B: Solving Equations by Using
Multiplication or Division
Solve the equation. Check your answer.
–4.8 = –6v
0.8 = v
Check
–4.8 = –6v
–4.8 –6(0.8)
–4.8
–4.8 
Since v is multiplied by –6,
divide both sides by –6 to
undo the multiplication.
The solution set is {0.8}.
To check your solution,
substitute 0.8 for v in the
original equation.
2-1 Solving One-Step Equations
Check It Out! Example 2a
Solve each equation. Check your answer.
p = 50
Check
Since p is divided by 5, multiply
both sides by 5 to undo the
division.
The solution set is {50}.
To check your solution,
substitute 50 for p in the
original equation.
10
10 
2-1 Solving One-Step Equations
Check It Out! Example 2b
Solve each equation. Check your answer.
0.5y = –10
y = –20
0.5y = –10
Check
0.5(–20) –10
–10
–10 
Since y is multiplied by 0.5,
divide both sides by 0.5 to
undo the multiplication.
The solution set is {–20}.
To check your solution,
substitute –20 for y in the
original equation.
2-1 Solving One-Step Equations
Check It Out! Example 2c
Solve each equation. Check your answer.
c = 56
Check
Since c is divided by 8,
multiply both sides by 8 to
undo the division.
The solution set is {56}.
To check your solution,
substitute 56 for c in the
original equation.
7
7 
2-1 Solving One-Step Equations
When solving equations, you will sometimes
find it easier to add an opposite to both
sides instead of subtracting or to multiply by
a reciprocal instead of dividing. This is often
true when an equation contains negative
numbers or fractions.
2-1 Solving One-Step Equations
Additional Example 3A: Solving Equations by Using
Opposites or Reciprocals
Solve each equation.
The reciprocal of
is
w is multiplied by
both sides by
. Since
multiply
.
The solution set is {–24}.
2-1 Solving One-Step Equations
Additional Example 3B: Solving Equations by Using
Opposites or Reciprocals
Solve each equation.
Since p is added to
, add
to both sides to undo the
subtraction.
The solution set is
.
{ }
2-1 Solving One-Step Equations
Check It Out! Example 3a
Solve the equation. Check your answer.
–2.3 + m = 7
–2.3 + m = 7
+2.3
+ 2.3
m = 9.3
Check –2.3 + m = 7
–2.3 + 9.3 7
7 7
Since –2.3 is added to m,
add 2.3 to both sides.
The solution set is {9.3}.
To check your solution,
substitute 9.3 for m in the
original equation.
2-1 Solving One-Step Equations
Check It Out! Example 3b
Solve the equation. Check your answer.
5
+z= 4
Since
is added to z add
to both sides.
The solution set is {2}.
Check
To check your solution,
substitute 2 for z in the
original equation.

2-1 Solving One-Step Equations
Check It Out! Example 3c
Solve the equation. Check your answer.
The reciprocal of
is
w is multiplied by
both sides by
w = 612
. Since
multiply
.
The solution set is {612}.
Check
102
102 
To check your solution,
substitute 612 for w in the
original equation.
2-1 Solving One-Step Equations
Additional Example 4: Application
Ciro deposits 1
of the money he earns from
4
mowing lawns into a college education fund. This
year Ciro added $285 to his college education
fund. Write and solve an equation to find out how
much money Ciro earned mowing lawns this year.
2-1 Solving One-Step Equations
Additional Example 4 Continued
1
4
times

earnings
e
is
$285
=
$285
Write an equation to represent the
relationship.
4
1
4
1  4 e = 1  285
e = $1140
1
4
The reciprocal of 4 is 1 . Since e
is multiplied by 1 ,
4
multiply both sides by 4 .
1
The original earnings were $1140 .
2-1 Solving One-Step Equations
Check It Out! Example 4
The distance in miles from the airport that a
plane should begin descending divided by 3
equals the plane’s height above the ground
in thousands of feet. A plane is 10,000 feet
above the ground. Write and solve an
equation to find the distance from the
airport at which this plane should begin
descending.
2-1 Solving One-Step Equations
Check It Out! Example 4 Continued
distance
divided by
3
is
d
÷
3
=
height
h
Write an equation to represent the
relationship.
3 d = 3 10
1 3
1
d = 30
Substitute 10 for h. The reciprocal of 13
is 3 . Since d is multiplied by 13
1
multiply both sides by 3 .
1
At 10,000 feet altitude the decent should start 30,000 feet
from the airport.
2-1 Solving One-Step Equations
Lesson Quiz
Solve each equation.
1. r – 4 = –8
–4
3.
5.
2.8
2.
4. 8y = 4
40
6. This year a high school had 578 sophomores
enrolled. This is 89 less than the number enrolled
last year. Write and solve an equation to find the
number of sophomores enrolled last year.
s – 89 = 578; s = 667