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2
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Guided Practice
Here there are two blue flies. You can write a lot of different equations to represent this situation.

Suppose d represents the leftmost blue fly, and j represents the rightmost fly.

You can write j = 9, d + 2 = 9, 9 – 2 = d, d + 2 = j,
j – 2 = d, and 9 – d = 2. All of these equations represent the situation.

The leftmost blue fly, d, is located at 7, because 9 – 2 = 7.

Tim's father is 3 times as old as Tim is. If his father is 39 years old, how old is Tim?

Tim's father is 39, which is 3 times Time's age, t.

3t = 39

Divide both sides by 3 to solve.

Tim is 13 years old.

Write the inequality or equation that is represented by the model. Use any variable you want.

Write the inequality or equation that is represented by the model. Use any variable you want.

Write the inequality or equation that is represented by the model. Use any variable you want.

Solve the equation.
Draw a number line model to help you.

  b + 5 = 10

   b =

Solve the equation.
Draw a number line model to help you.

  b – 5 = 10

   b =

Solve the equation.
Draw a number line model to help you.

  1 + c = 8

   c =

Solve the equation.
Draw a number line model to help you.

  4 + x = 4

   x =

Solve the equation.
Draw a number line model to help you.

  k + 6 = 6 + 5

   k =

Solve the equation.
Draw a number line model to help you.

  3 + 7 = k + 3

   k =

Solve the equation.
Draw a number line model to help you.

  7 + (m + 2) = (7 + 7) + 2

   m =

Sergio has collected 48 pennies, which is double the number Tom has collected. How many has Tom collected? Write an equation with a variable. Then solve the equation.

× =

+ =

Mario is 2.5 inches shorter than his sister. If his sister is 60.5 inches tall, how tall is Mario? Write an equation with a variable. Then solve the equation.

+ =

+ =

Mario is 2.5 inches taller than his sister. If his sister is 60.5 inches tall, how tall is Mario? Write an equation with a variable. Then solve the equation.

=

+ =

Today, 2/3 of Robin's friends have cell phones.

If 80 of her friends have cell phones, how many friends does she have? Write an equation with a variable. Then solve.

× =

× =

You can build an equation using data from a table.

Suppose that a car travels at 65 miles per hour. The table shows the distance in miles the car traveled, d, based on the time, t, in hours.

Time (h)Distance (mi)
00
165
2130
3195
td = 65t

The equation d = 65t describes the relationship between the distance traveled and time in this situation.

Sometimes quantities are related to each other such that one quantity depends on the other.

For example, if a car travels at 65 miles per hour, the number of miles the car goes is dependent on how many hours have passed.

Time (h)Distance (mi)
00
165
2130
3195
td = 65t

Miles is the dependent quantity and hours is the independent quantity.

A car travels at 45 miles per hour. The table shows the distance in miles traveled, d, based on the time, t, in hours.

Write the equation for the distance, d.

Time (h)Distance (mi)
00
145
290
3135
t?

d =

The table shows the number of quarters, q, in change you receive from a machine based on the number of dollars, d, you put in.

Write the equation for the number of quarters.

DollarsQuarters
00
14
28
312
d?

q =

The table shows the number of quarters, q, in change you receive from a machine based on the number of dollars, d, you put in.

Write the equation for the number of dollars using a decimal.

DollarsQuarters
00
14
28
312
?q

d =

The table shows the amount of gas in gallons, g, and the cost, c, of the gas in dollars.

Write the equation for the cost of the gas.

GallonsCost ($)
00
12.83
25.66
38.49
g?

c =

The table shows the number of pounds of potatoes, p, and the cost of the potatoes, c.

Write the equation for the cost of the potatoes.

PoundsCost ($)
00
11.19
22.38
33.57
p?

c =

The table shows the number of ounces of movie popcorn, p, and the cost, c.

Write the equation for the cost of the popcorn.

OuncesCost ($)
12.50
22.50
32.50
42.50
p?

c =

The table shows the number of rides, r, you can get for a given number of tickets, t.

Write the equation for the number of rides. (Use a decimal.)

TicketsRides
00
41
82
123
t?

r =

The table shows the number of miles, m, that Brad walks in a given number of hours, h.

Write the equation for the miles walked.

HoursMiles
00
15
210
315
h?

m =

A car travels at 45 miles per hour. The table shows the distance in miles traveled, d, based on the time, t, in hours.

Write the independent quantity and dependent quantity for this situation.

Time (h)Distance (mi)
00
145
290
3135
t?

independent quantity:

independent quantity:

The table shows the number of quarters, q, in change you receive from a machine based on the number of dollars, d, you put in.

Write the independent quantity and dependent quantity for this situation.

DollarsQuarters
00
14
28
312
d?

independent quantity:

independent quantity:

The table shows the cost of gas, c, for each amount in gallons, g, that you put into your car.

Write the independent quantity and dependent quantity for this situation.

GallonsCost ($)
00
12.83
25.66
38.49
g?

independent quantity:

independent quantity:

The table shows the number of gallons of gas, g, for each amount of money, m, in dollars, that you spend on gas.

Write the independent quantity and dependent quantity for this situation.

Amount ($)Gallons
00
2.831
5.662
8.493
m?

independent quantity:

independent quantity:

The table shows the cost, c, of different pounds of potatoes, p, that you buy.

Write the independent quantity and dependent quantity for this situation.

PoundsCost ($)
00
11.19
22.38
33.57
p?

independent quantity:

independent quantity:

The table shows the number of rides, r, you can get for a given number of tickets, t.

Write the independent quantity and dependent quantity for this situation.

TicketsRides
00
41
82
123
t?

independent quantity:

independent quantity:

The table shows how far Brad walks in a given amount of time.

Write the independent quantity and dependent quantity for this situation.

Time (h)Distance (mi)
00
15
210
315
h?

independent quantity:

independent quantity:

Mathematically, solving an equation means to decide which value from a set makes the equation true.

Consider this equation: x + 2 = 10. Which set contains the value of x?

A : {2, 4, 6, 8, 10, …}

B : {0, –1, –2, –3, –4, …}

The value for x that makes the equation true is 8. So, Set A contains the solution to the equation. Set B does not contain the value 8. It contains all the integers less than or equal to 0.

Which set contains the solution to the equation?

p + 4 = 4

A : {2, 4, 6, 8, 10, …}

B : {0, –1, –2, –3, –4, …}

Set contains the solution to the equation, which is p = .

Which set contains the solution to the equation?

p + 4 = 4

A : {… 2, 4, 6, 8, 10, …}

B : {–1, –2, –3, –4, …}

Set contains the solution to the equation, which is p = .

Which set contains the solution to the equation?

2m = 12

A : {1, 2, 3, 4, …}

B : {1, 3, 5, 7, …}

Set contains the solution to the equation, which is m = .

Which set contains the solution to the equation?

2/3q = 48

A : {–1, –2, –3, –4, …}

B : {1, 2, 3, 4, …}

Set contains the solution to the equation, which is q = .

Which set contains the solution to the equation?

25 = 115 – x

A : {1, 3, 5, 7, …}

B : {10, 20, 30, 40, …}

Set contains the solution to the equation, which is x = .

Which set contains the solution to the equation?

144 = 72/n

A : {1/2, 1, 3/2, 2, 5/2, …}

B : {1/3, 2/3, 1, 4/3, 5/3, …}

Set contains the solution to the equation, which is n = .

Which set contains the solution to the equation?

r = –10/1

A : {–1, –2, –3, –4, …}

B : {0}

Set contains the solution to the equation, which is r = .

=

You can enter numbers, lowercase variables, or addition expressions into each side of the interactive balance on the left—expressions like 6, p, r + 2, or 3x + 1. Then press Enter to show the expression on the balance. No subtraction expressions or parentheses, though.

Coefficients (the numbers that are multiplied to variables; like the 6 in 6x) can be whole numbers from 0 to 6, and constants can be whole numbers from 0 to 6.

Try this. What value for the variable x makes the equation 3x + 1 = 6 + 4 balance? Highlight an x on the balance and type a number to replace it. Then see if you're right.

x =

Use the balance to determine the solution of the equation b + 4 = 6 + 2.

b =

Use the balance to determine the solution of the equation 4b + 3 = 6 + 5.

b =

Use the balance to determine the solution of the equation 5 + 4 = 2h + 1.

h =

Use the balance to determine the solution of the equation 6 + 2 = 5w + 3.

w =

Use the balance to determine the solution of the equation 4k = 6 + 6.

k =

The inequality symbol > means "is greater than." The inequality symbol < means "is less than." The equality symbol = means "is equal to."

Your turn. Write two other inequalities and an equation to represent the models.

> 3

< q

= x

Write two inequalities and an equation to represent the models.

<

>

=

Write two inequalities and an equation to represent the models.

<

>

=

Write two inequalities and an equation to represent the models.

<

>

=

Write two inequalities and an equation to represent the models.

<

>

=

Write two inequalities and an equation to represent the models.

<

>

=

Write two inequalities and an equation to represent the models.

<

>

=

Write two inequalities and an equation to represent the model.

<

>

=

Your turn. Solve the equation x + 10 = 15.

x =

Solve each equation.

x + 5 = 8      x =

5 + x = 8      x =

8 = x + 5      x =

8 = 5 + x      x =

Solve each equation.

x + 0 = 4      x  =

y + 0 = 9      y =

0 + b = 3      b =

7 = 0 + c      c =  

Solve each equation.

12 = 12 • m      m =

40 = 40 • n      n =

s • 32 = 32      s =  

27 • q = 27      q =  

Solve each equation.

t + 5 = 5 + 9       t =   

b × 2 = 2 × 3      b =  

4 × 8 = w × 4      w =

6 + a = 5 + 6      a =  

Solve each equation.

     j/5 = 2/5      j =

   j ÷ 5 = 2/5      j =

 1/5× j = 2/5      j =

Solve each equation.

m ÷ 3 = 6      m =

m × 3 = 6      m =

m – 3 = 6      m =

m + 3 = 6      m =

Your turn. Write an equation to model the situation. Then solve the equation. Use any letter for the variable. Use +, -, *, and /.

Brett's dad is 40 years older than Brett. If his dad is 48 years old, how old is Brett?

Equation:
=

Brett is years old.

Write an equation to model the situation. Then solve the equation. Use any letter for the variable. Use +, -, *, and /.

May has 14 more pennies than nickels. If she has 82 nickels, how many pennies does she have?

Equation:
=

May has pennies.

Write an equation to model the situation. Then solve the equation. Use any letter for the variable. Use +, -, *, and /.

May has twice as many pennies as nickels. If she has 82 nickels, how many pennies does she have?

Equation:
=

May has pennies.

Write an equation to model the situation. Then solve the equation. Use any letter for the variable. Use +, -, *, and /.

May has half as many pennies as nickels. If she has 82 nickels, how many pennies does she have?

Equation:
=

May has pennies.

Write an equation to model the situation. Then solve the equation. Use any letter for the variable. Use +, -, *, and /.

Sam finished the race in 229 seconds. It took him 1/3 as long to finish as Nellie. How long did it take Nellie to finish the race?

Equation:
=

Nellie finished in seconds.

Write an equation to model the situation. Then solve the equation. Use any letter for the variable. Use +, -, *, and /.

Sam finished the race in 229 seconds. It took him 35 seconds longer than Clint to finish. How long did it take Clint to finish the race?

Equation:
=

Clint finished in seconds.

Write an equation to model the situation. Then solve the equation. Use any letter for the variable. Use +, -, *, and /.

Sam finished the race in 229 seconds. It took him 109 seconds less than Dan to finish. How long did it take Dan to finish the race?

Equation:
=

Dan finished in seconds.