Download chapter 7_8

Solving Equations with
7-8 Variables on Both Sides
Warm Up
Problem of the Day
Lesson Presentation
Lesson Quizzes
Solving Equations with
7-8 Variables on Both Sides
Warm Up
Solve.
1. 2x + 9x – 3x + 8 = 16
x=1
2. –4 = 6x + 22 – 4x x = –13
3. 2 + x = 5 1 x = 34
7 7
7
4. 9x – 2x = 3 1
16 4
8
x = 50
Solving Equations with
7-8 Variables on Both Sides
Problem of the Day
An equilateral triangle and a regular
pentagon have the same perimeter.
Each side of the pentagon is 3 inches
shorter than each side of the triangle.
What is the perimeter of the triangle?
22.5 in.
Solving Equations with
7-8 Variables on Both Sides
Sunshine State Standards
Preview of MA.912.A.3.1 Solve linear
equations in one variable that include
simplifying algebraic expressions.
Solving Equations with
7-8 Variables on Both Sides
Some problems produce equations that have
variables on both sides of the equal sign.
Solving an equation with variables on both
sides is similar to solving an equation with a
variable on only one side. You can add or
subtract a term containing a variable on both
sides of an equation.
Solving Equations with
7-8 Variables on Both Sides
Additional Example 1A: Solving Equations with
Variables on Both Sides
Solve.
4x + 6 = x
4x + 6 = x
– 4x
– 4x
6 = –3x
6 = –3x
–3
–3
–2 = x
Subtract 4x from both sides.
Divide both sides by –3.
Solving Equations with
7-8 Variables on Both Sides
Helpful Hint
Check your solution by substituting the value
back into the original equation. For example,
4(-2) + 6 = -2 or -2 = -2.
Solving Equations with
7-8 Variables on Both Sides
Additional Example 1B: Solving Equations with
Variables on Both Sides
Solve.
9b – 6 = 5b + 18
9b – 6 = 5b + 18
– 5b
– 5b
Subtract 5b from both sides.
4b – 6 = 18
+6 +6
4b = 24
4b = 24
4
4
b=6
Add 6 to both sides.
Divide both sides by 4.
Solving Equations with
7-8 Variables on Both Sides
Additional Example 1C: Solving Equations with
Variables on Both Sides
Solve.
9w + 3 = 9w + 7
9w + 3 = 9w + 7
– 9w
– 9w
3≠
Subtract 9w from both sides.
7
No solution. There is no number that can be
substituted for the variable w to make the
equation true.
Solving Equations with
7-8 Variables on Both Sides
Helpful Hint
If the variables in an equation are eliminated
and the resulting statement is false, the
equation has no solution.
Solving Equations with
7-8 Variables on Both Sides
Check It Out: Example 1A
Solve.
5x + 8 = x
5x + 8 = x
– 5x
– 5x
8 = –4x
8 = –4x
–4
–4
–2 = x
Subtract 5x from both sides.
Divide both sides by –4.
Solving Equations with
7-8 Variables on Both Sides
Check It Out: Example 1B
Solve.
3b – 2 = 2b + 12
3b – 2 = 2b + 12
– 2b
– 2b
Subtract 2b from both sides.
b–2=
+2
b
=
12
+ 2 Add 2 to both sides.
14
Solving Equations with
7-8 Variables on Both Sides
Check It Out: Example 1C
Solve.
3w + 1 = 3w + 8
3w + 1 = 3w + 8
– 3w
– 3w
1≠
Subtract 3w from both sides.
8
No solution. There is no number that can be
substituted for the variable w to make the
equation true.
Solving Equations with
7-8 Variables on Both Sides
To solve multi-step equations with variables on
both sides, first combine like terms and clear
fractions. Then add or subtract variable terms
to both sides so that the variable occurs on
only one side of the equation. Then use
properties of equality to isolate the variable.
Solving Equations with
7-8 Variables on Both Sides
Additional Example 2: Solving Multi-Step Equations
with Variables on Both Sides
Solve.
10z – 15 – 4z = 8 – 2z - 15
10z – 15 – 4z = 8 – 2z – 15
6z – 15 = –2z – 7 Combine like terms.
+ 2z
+ 2z
Add 2z to both sides.
8z – 15
+ 15
8z
8z
8
z
=
=8
= 8
8
=1
–7
+15 Add 15 to both sides.
Divide both sides by 8.
Solving Equations with
7-8 Variables on Both Sides
Check It Out: Example 2
Solve.
12z – 12 – 4z = 6 – 2z + 32
12z – 12 – 4z = 6 – 2z + 32
8z – 12 = –2z + 38 Combine like terms.
+ 2z
+ 2z
Add 2z to both sides.
10z – 12 =
38
+ 12
+12 Add 12 to both sides.
10z = 50
10z = 50
Divide both sides by 10.
10
10
z=5
Solving Equations with
7-8 Variables on Both Sides
Additional Example 3: Business Application
Daisy’s Flowers sell a rose bouquet for
$39.95 plus $2.95 for every rose. A
competing florist sells a similar bouquet
for $26.00 plus $4.50 for every rose. Find
the number of roses that would make both
florists’ bouquets cost the same price.
Solving Equations with
7-8 Variables on Both Sides
Additional Example 3 Continued
39.95 + 2.95r = 26.00 + 4.50r
– 2.95r
39.95
– 2.95r
=
– 26.00
13.95
Subtract 2.95r from
both sides.
26.00 + 1.55r
Subtract 26.00 from
both sides.
– 26.00
=
Let r represent the
price of one rose.
1.55r
13.95
1.55r
Divide both sides by
=
1.55
1.55
1.55.
9=r
The two services would cost the same when
purchasing 9 roses.
Solving Equations with
7-8 Variables on Both Sides
Check It Out: Example 3
Marla’s Gift Baskets sells a muffin basket
for $22.00 plus $2.25 for every balloon. A
competing service sells a similar muffin
basket for $16.00 plus $3.00 for every
balloon. Find the number of balloons that
would make both gift basket companies
muffin baskets cost the same price.
Solving Equations with
7-8 Variables on Both Sides
Check It Out: Example 3 Continued
22.00 + 2.25b = 16.00 + 3.00b
– 2.25b
22.00
– 2.25b
Let b represent the
price of one balloon.
Subtract 2.25b from
both sides.
= 16.00 + 0.75b
– 16.00
– 16.00
6.00
=
6.00
0.75
0.75b
Subtract 16.00 from
both sides.
0.75b
Divide both sides by
0.75
0.75.
8=b
The two services would cost the same when purchasing
8 balloons.
=
Solving Equations with
7-8 Variables on Both Sides
Lesson Quizzes
Standard Lesson Quiz
Lesson Quiz for Student Response Systems
Solving Equations with
7-8 Variables on Both Sides
Lesson Quiz: Part I
Solve.
1. 4x + 16 = 2x
x = –8
2. 8x – 3 = 15 + 5x
x=6
3. 2(3x + 11) = 6x + 4 no solution
1
1
x = 36
4. 4 x = 2 x – 9
5. An apple has about 30 calories more than an
orange. Five oranges have about as many calories
as 3 apples. How many calories are in each?
An orange has 45 calories. An apple has 75 calories.
Solving Equations with
7-8 Variables on Both Sides
Lesson Quiz: Part II
5. The equation S = 2B + 2rh gives the
surface area S of a cylinder with base area
B, radius r, and height h. Solve the equation
for h.
Solving Equations with
7-8 Variables on Both Sides
Lesson Quiz for Student Response Systems
1. Combine like terms. 4p + 14 = 11p
A. p = 2
B. p = 7
C. p = 14
D. p = 15
Solving Equations with
7-8 Variables on Both Sides
Lesson Quiz for Student Response Systems
2. Combine like terms. 3g – 6 = 4g – 7
A. g = 1
B. g = 7
C. g = 13
D. g = –13
Solving Equations with
7-8 Variables on Both Sides
Lesson Quiz for Student Response Systems
3. Combine like terms. 3(g – 2) = 7g – 18
A. g = 2
B. g = 3
C. g = –3
D. g = –2
Solving Equations with
7-8 Variables on Both Sides
Lesson Quiz for Student Response Systems
4. Solve for D.
A.
B. D = V
C. D = mV
D.