Download 2-4 Solving Equations with Variables on Both Sides

2-4
Solving Equations with Variables on Both Sides
Preview
Warm Up
California Standards
Lesson Presentation
2-4
Solving Equations with Variables on Both Sides
Warm Up
Simplify.
1. 4x – 10x
2. –7(x – 3)
–6x
–7x + 21
3.
2x + 3
4. 15 – (x – 2) 17 – x
Solve.
5. 3x + 2 = 8 2
6.
28
2-4
Solving Equations with Variables on Both Sides
California
Standards
4.0 Students simplify expressions before
solving linear equations and inequalities in one
variable, such as 3(2x – 5) + 4(x – 2) = 12.
5.0 Students solve multistep problems,
including word problems, involving linear
equations and linear inequalities in one variable
and provide justification for each step.
2-4
Solving Equations with Variables on Both Sides
Vocabulary
identity
2-4
Solving Equations with Variables on Both Sides
To solve an equation with variables on both sides,
use inverse operations to "collect" variable terms
on one side of the equation.
Helpful Hint
Equations are often easier to solve when the
variable has a positive coefficient. Keep this in
mind when deciding on which side to "collect"
variable terms.
2-4
Solving Equations with Variables on Both Sides
Additional Example 1: Solving Equations with
Variables on Both Sides
Solve 7n – 2 = 5n + 6.
7n – 2 = 5n + 6
–5n
–5n
2n – 2 =
+2
2n
=
6
+2
8
To collect the variable terms on one
side, subtract 5n from both sides.
Since 2 is subtracted from 2n, add 2
to both sides.
Since n is multiplied by 2, divide both
sides by 2 to undo the multiplication.
n=4
2-4
Solving Equations with Variables on Both Sides
Check It Out! Example 1a
Solve the equation. Check your answer.
4b + 2 = 3b
4b + 2 = 3b
–3b
–3b
b+2= 0
–2 –2
b = –2
To collect the variable terms on
one side, subtract 3b from
both sides.
Since 2 is added to b, subtract
2 from both sides.
2-4
Solving Equations with Variables on Both Sides
Check It Out! Example 1a Continued
Solve the equation. Check your answer.
Check
4b + 2 = 3b
4(–2) + 2
–8 + 2
–6
3(–2)
–6
–6 
To check your answer,
substitute –2 for b.
2-4
Solving Equations with Variables on Both Sides
Check It Out! Example 1b
Solve the equation. Check your answer.
0.5 + 0.3y = 0.7y – 0.3
0.5 + 0.3y = 0.7y – 0.3
–0.3y –0.3y
0.5
= 0.4y – 0.3
+0.3
+ 0.3
0.8
= 0.4y
2=y
To collect the variable terms
on one side, subtract 0.3y
from both sides.
Since 0.3 is subtracted from
0.4y, add 0.3 to both sides
to undo the subtraction.
Since y is multiplied by 0.4,
divide both sides by 0.4 to
undo the multiplication.
2-4
Solving Equations with Variables on Both Sides
Check It Out! Example 1b Continued
Solve the equation. Check your answer.
Check 0.5 + 0.3y = 0.7y – 0.3
0.5 + 0.3(2) 0.7(2) – 0.3
0.5 + 0.6 1.4 – 0.3
1.1 1.1 
To check your answer,
substitute 2 for y.
2-4
Solving Equations with Variables on Both Sides
To solve more complicated
equations, you may need to first
simplify by using the Distributive
Property or combining like terms.
2-4
Solving Equations with Variables on Both Sides
Additional Example 2: Simplifying Each Side Before
Solving Equations
Solve the equation.
4 – 6a + 4a = –1 – 5(7 – 2a)
4 – 6a + 4a = –1 –5(7 – 2a)
Distribute –5 to the
expression in
4 – 6a + 4a = –1 –5(7) –5(–2a)
parentheses.
4 – 6a + 4a = –1 – 35 + 10a
Combine like terms.
4 – 2a = –36 + 10a
Since –36 is added to
+36
+36
10a, add 36 to both
40 – 2a =
10a
sides.
2-4
Solving Equations with Variables on Both Sides
Additional Example 2 Continued
Solve 4 – 6a + 4a = –1 – 5(7 – 2a).
40 – 2a = 10a
+ 2a +2a
40
= 12a
To collect the variable terms
on one side, add 2a to both
sides.
Since a is multiplied by 12,
divide both sides by 12.
2-4
Solving Equations with Variables on Both Sides
Check It Out! Example 2a
Solve the equation. Check your answer.
1
Distribute to the expression in
2
parentheses.
3=b–1
+1
+1
4=b
To collect the variable terms on
one side, subtract 1 b from
2
both sides.
Since 1 is subtracted from
b, add 1 to both sides.
2-4
Solving Equations with Variables on Both Sides
Check It Out! Example 2a Continued
Solve the equation. Check your answer.
Check
To check your answer,
substitute 4 for b.
5
5

2-4
Solving Equations with Variables on Both Sides
Check It Out! Example 2b
Solve the equation. Check your answer.
3x + 15 – 9 = 2(x + 2)
3x + 15 – 9 = 2(x + 2)
3x + 15 – 9 = 2(x) + 2(2)
3x + 15 – 9 = 2x + 4
3x + 6 = 2x + 4
–2x
–2x
x+6=
4
–6
–6
x = –2
Distribute 2 to the expression
in parentheses.
Combine like terms.
To collect the variable terms
on one side, subtract 2x
from both sides.
Since 6 is added to x, subtract
6 from both sides.
2-4
Solving Equations with Variables on Both Sides
Check It Out! Example 2b Continued
Solve the equation. Check your answer.
Check 3x + 15 – 9 = 2(x + 2)
3(–2) + 15 – 9
–6 + 15 – 9
0
2(–2 + 2) To check your answer,
substitute –2 for x.
2(0)
0
2-4
Solving Equations with Variables on Both Sides
An identity is an equation that is always
true, no matter what value is substituted
for the variable. The solution set of an
identity is all real numbers. Some
equations are always false. Their solution
sets are empty. In other words, their
solution sets contain no elements.
2-4
Solving Equations with Variables on Both Sides
Additional Example 3A: Infinitely Many Solutions or
No Solutions
Solve the equation.
10 – 5x + 1 = 7x + 11 – 12x
10 – 5x + 1 = 7x + 11 – 12x
11 – 5x = 11 – 5x
Identify like terms.
Combine like terms on
the left and the right.
The statement 11 – 5x = 11 – 5x is true for all values
of x. The equation 10 – 5x + 1 = 7x + 11 – 12x is an
identity. All values of x will make the equation true.
In other words, all real numbers are solutions.
2-4
Solving Equations with Variables on Both Sides
Additional Example 3B: Infinitely Many Solutions or
No Solutions
Solve the equation.
12x – 3 + x = 5x – 4 + 8x
12x – 3 + x = 5x – 4 + 8x Identify like terms.
Combine like terms on the left
13x – 3 = 13x – 4
and the right.
–13x
–13x
Subtract 13x from both sides.
–3 = –4 
False statement; the solution
set is .
The equation 12x – 3 + x = 5x – 4 + 8x is always
false. There is no value of x that will make the
equation true. There are no solutions.
2-4
Solving Equations with Variables on Both Sides
Writing Math
The empty set can be written as  or {}.
2-4
Solving Equations with Variables on Both Sides
Check It Out! Example 3a
Solve the equation.
4y + 7 – y = 10 + 3y
4y + 7 – y = 10 + 3y
Identify like terms.
3y + 7 = 3y + 10
–3y
–3y
7 = 10
Subtract 3y from both sides.
False statement; the

solution set is .
The equation 4y + 7 – y = 10 + 3y is always
false. There is no value of y that will make the
equation true. There are no solutions.
2-4
Solving Equations with Variables on Both Sides
Check It Out! Example 3b
Solve the equation.
2c + 7 + c = –14 + 3c + 21
2c + 7 + c = –14 + 3c + 21
3c + 7 = 3c + 7
Identify like terms.
Combine like terms on the
left and the right.
The statement 3c + 7 = 3c + 7 is true for all values
of c. The equation 2c + 7 + c = –14 + 3c + 21 is an
identity. All values of c will make the equation true.
In other words, all real numbers are solutions.
2-4
Solving Equations with Variables on Both Sides
Additional Example 4: Application
Jon and Sara are planting tulip bulbs. Jon has
planted 60 bulbs and is planting at a rate of 44
bulbs per hour. Sara has planted 96 bulbs and
is planting at a rate of 32 bulbs per hour. In
how many hours will Jon and Sara have planted
the same number of bulbs? How many bulbs
will that be?
Person
Bulbs
Jon
60 bulbs plus 44 bulbs per hour
Sara
96 bulbs plus 32 bulbs per hour
2-4
Solving Equations with Variables on Both Sides
Additional Example 4 Continued
Let h represent hours, and write expressions for
the number of bulbs planted.
When is
60
60
plus
bulbs
+
44h
44
bulbs
each
hour
=
the
same
as
96
96
bulbs
+
plus
32
bulbs
?
each
hour
32h
60 + 44h = 96 + 32h To collect the variable terms
on one side, subtract 32h
–32h
–32h
from both sides.
60 + 12h = 96
2-4
Solving Equations with Variables on Both Sides
Additional Example 4 Continued
60 + 12h = 96
–60
– 60
12h = 36
h=3
Since 60 is added to 12h,
subtract 60 from both sides.
Since h is multiplied by 12,
divide both sides by 12 to
undo the multiplication.
2-4
Solving Equations with Variables on Both Sides
Additional Example 4 Continued
After 3 hours, Jon and Sara will have planted the
same number of bulbs. To find how many bulbs they
will have planted in 3 hours, evaluate either
expression for h = 3:
60 + 44h = 60 + 44(3) = 60 + 132 = 192
96 + 32h = 96 + 32(3) = 96 + 96 = 192
After 3 hours, Jon and Sara will each have planted
192 bulbs.
2-4
Solving Equations with Variables on Both Sides
Check It Out! Example 4
Four times Greg's age, decreased by 3 is equal
to 3 times Greg's age, increased by 7. How old
is Greg?
Let g represent Greg's age, and write expressions
for his age.
Four
times
Greg's
age
4g
decreased
by
3
–
3
is
equal
to
=
three
times
Greg's
age
3g
increased
by
+
7
7
.
2-4
Solving Equations with Variables on Both Sides
Check It Out! Example 4 Continued
4g – 3 = 3g + 7
–3g
–3g
g–3=
+3
g=
7
+3
10
Greg is 10 years old.
To collect the variable terms on
one side, subtract 3g from both
sides.
Since 3 is subtracted from g, add
3 to both sides.
2-4
Solving Equations with Variables on Both Sides
Lesson Quiz
Solve each equation.
1. 7x + 2 = 5x + 8 3
2. 4(2x – 5) = 5x + 4 8
3. 6 – 7(a + 1) = –3(2 – a)
4. 4(3x + 1) – 7x = 6 + 5x – 2 all real numbers
5.
1
20 hours
6. A painting company charges $250 base plus $16
per hour. Another painting company charges $210
base plus $18 per hour. How long is a job for
which the two companies costs are the same?