Understanding Equations: Variables, Solutions to an Equation, and Mathematical Modeling

Value of $x$	Substitute	Are both sides equal?
$5$	$5 - 5 = ? 3$	$0 \neq = 3 \times$
$6$	$6 - 5 = ? 3$	$1 \neq = 3 \times$
$7$	$7 - 5 = ? 3$	$2 \neq = 3 \times$
$8$	$8 - 5 = ? 3$	$3 = 3 ✓$

Value of

x

Substitute

Are both sides equal?

5

5 - 5 = ? 3

0 \neq = 3 \times

6

6 - 5 = ? 3

1 \neq = 3 \times

7

7 - 5 = ? 3

2 \neq = 3 \times

8

8 - 5 = ? 3

3 = 3 ✓

Menu	Menu Price	Substitution into $2 x + 8 = 22$	Check
Combo I	$7$	$2 (7) + 8 = ? 22$	$22 = 22 ✓$
Combo II	$9$	$2 (9) + 8 = ? 22$	$26 \neq = 22 \times$
Combo III	$5$	$2 (5) + 8 = ? 22$	$18 \neq = 22 \times$
Combo IV	$6$	$2 (6) + 8 = ? 22$	$20 \neq = 22 \times$

Menu Price

Substitution into

2 x + 8 = 22

Check

Combo I

7

2 (7) + 8 = ? 22

22 = 22 ✓

Combo II

9

2 (9) + 8 = ? 22

26 \neq = 22 \times

Combo III

5

2 (5) + 8 = ? 22

18 \neq = 22 \times

Combo IV

6

2 (6) + 8 = ? 22

20 \neq = 22 \times

Menu	Menu Price	Substitution into $x^{2} - 15 x = - 54$	Check
Combo I	$7$	$(7)^{2} - 15 (7) = ? - 54 49 - 105 = ? - 54$	$- 56 \neq = - 54 \times$
Combo II	$9$	$(9)^{2} - 15 (9) = ? - 54 81 - 135 = ? - 54$	$- 54 = - 54 ✓$
Combo III	$5$	$(5)^{2} - 15 (5) = ? - 54 25 - 75 = ? - 54$	$- 50 \neq = - 54 \times$
Combo IV	$6$	$(6)^{2} - 15 (6) = ? - 54 36 - 90 = ? - 54$	$- 54 = - 54 ✓$

Menu Price

Substitution into

x^{2} - 15 x = - 54

Check

Combo I

7

(7)^{2} - 15 (7) = ? - 54 49 - 105 = ? - 54

- 56 \neq = - 54 \times

Combo II

9

(9)^{2} - 15 (9) = ? - 54 81 - 135 = ? - 54

- 54 = - 54 ✓

Combo III

5

(5)^{2} - 15 (5) = ? - 54 25 - 75 = ? - 54

- 50 \neq = - 54 \times

Combo IV

6

(6)^{2} - 15 (6) = ? - 54 36 - 90 = ? - 54

- 54 = - 54 ✓

Write the given sentence as an equation.

5 less than a number b equals 9 .

The sum of a number y and 6 is 13 .

When writing words as equations, we look for keywords such as is or equals to find the placement of the equals sign. 5 less than a number b equals 9. ↓ = Another keyword in our sentence is less than, which indicates subtraction. 5 less than a number b equals 9. ⇓ ↓ - = Now let's rewrite the other terms 5, the number b, and 9 to complete our equation.

The equation b-5=9 represents the given sentence.

Let's start by examining the given sentence again. Note that the word sum means addition. The sum of a number y and 3 is 13. ⇓ + Next, we place the equals sign where the keyword is stands. The sum of a number y and 6 is 13. ⇓ ⇓ + = Finally write the terms y, 6, and 13 to complete the equation.

The equation y+6=13 represents the given word sentence.

Consider the following equation.

x - 7 = 31

Choose the sentence that does not represent this equation.

We will decide which option does not represent the given equation. Let's start by examining the given options.

Sentences
7 subtracted from a number is 31.
A number decreased by 7 is 31.
A number minus 7 is the same as 31.
A number is 7 less than 31.
31 equals the difference between a number and 7.

Let's write each sentence as an equation and check whether it matches the given equation. When writing sentences as equations, we look for keywords such as is or equals to find the placement of the equals sign. There are also some other keywords to decide the place of the operational signs. In this case, we want to use subtraction, so let's look for keywords that indicate subtraction. difference less than subtracted from decreased by Since all of the options use the variable x, we will also use x to indicate a number. With this in mind, let's write one of the options as an equation.

We can write the other sentences as equations by following the same steps.

Sentences	Equation
7 subtracted from a number is 31.	x - 7 = 31
A number decreased by 7 is 31.	x - 7 = 31
A number minus 7 is the same as 31.	x - 7 = 31
A number is 7 less than 31.	x = 31 - 7
31 equals the difference between a number and 7.	31 = x - 7

As we can see from the table, the sentence A number is 7 less than 31. needs to be represented with a different equation. x=31-7 is not the same as x-7=31

A group of $6$ friends are planning to have a fun day together. They will choose one out of three different activities. The equations in the table represent the number of people $x$ who can participate in each activity for $$ 18 .$

Activity	Cost $($)$
Balloon Ride	$3 x - 42 = 18$
Amusement Park	$5 x - 22 = 18$
Zoo Trip	$2 x + 6 = 18$

Which activity should the friends choose if they want to spend exactly

$ 18

and all do the same activity?

We are asked to determine which activity the friends should choose if they want to spend exactly $18 and all do the same activity. There are 6 people in the friends group, so let's substitute x=6 into each equation and see which of them produce true statements. Let's start with the equation for the balloon ride.

Notice that x=6 is not a solution to this equation, so the 6 friends cannot join a balloon ride for $18. Let's apply the same thought process and substitute x=6 into the equations for the amusement park and zoo trip.

Activity	Cost ($)	Substitute x=6	Multiply	Add or Subtract	Is x=6 a Solution?
Balloon Ride	3x-42=18	3( 6)-42? =18	18-42? =18	-24=18 *	No
Amusement Park	5x-22=18	5( 6)-22? =18	30-22? =18	8 ≠ 18 *	No
Zoo Trip	2x+6=18	2( 6)+6? =18	12+6? =18	18=18 ✓	Yes

We found that x=6 is a solution to the equation that represents the cost of the zoo trip. This means that if the friends want to spend exactly $18 and all do something together, they should choose the zoo trip.

A scale balances with four violet blocks on one side and one orange

x -

block on the other side. This situation is represented by the following equation.

4 = x

Heichi places

16

more violet blocks to the left side. The scale is now unbalanced.

Scale with 20 violet blocks and one orange block.

In an attempt to balance the scale again, he placed more orange

x -

blocks on the right side and wrote the following equation that represents the balanced scale.

20 = ? \cdot x

Find the number that makes the equation true.

We know that the following scale is balanced when four violet blocks are on the left side and one orange x-block on the right side.

We want to find a number that balances the scale again when multiplied by x.

Let's keep adding 4 violet blocks to the left side and 1 orange x-block to the other side until we get 20 violet blocks. This method will keep the scale balanced!

As we can see, if there are 20 violet blocks on the left side of the scale, we need 5 orange x-blocks to keep the scale balanced. This means that we need to multiply the x-blocks by 5. Now we can complete the given equation. 20=5* x

Equations

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Finding the Correct Floor

Equation

Solution of an Equation

Extra

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Solution

Checking Solutions

Writing Equations

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Solution

Buying Games

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Solution

Selling Old Toys

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Solution

Finding the Correct Floor

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Solution

Equations

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