smallhead BIGWISDOM

Rising your being

Math – 6th Grade

Equations with first-degree variables

Exercise 1

On Monday, Emma had 5 apples; on Tuesday, she bought more apples to have a total of 12 apples. How many apples did she buy on Tuesday?

$\fontsize{15}{10}\selectfont \begin{flushleft} \raggedright \textbf{Solution:} \\[2ex]\textbf{Step 1: Define the variable.} \\Let $x$ be the number of apples Emma bought on Tuesday. \\[2ex] \textbf{Step 2: Write the equation.} \\[2ex]$ 5 + x = 12 $ \textit{\textcolor[HTML]{0000FF}{(5 apples that Emma had on Monday, plus $x$ apples she bought on Tuesday equals 12.)}}\\[2ex] % the [HTML]{0000FF} is for the blue color \textbf{Step 3: Solve the equation.} \\[2ex]Since the purpose is to find out the value of $x$, we need to isolate it. To do this, subtract 5 from both sides of the equation. According to the property of equality, "any operation performed on one side of the equation must also be performed on the other side." \\[2ex]$5 - 5 + x = 12 - 5$ \\[2ex]$0 + x = 7$ \\[2ex]$x = 7$ \\[2ex] \textbf{Step 4: Answer.} \\[2ex]\colorbox{yellow}{Emma bought $7$ apples on Tuesday.} \\[2ex] \textbf{Step 5: Verify.} \\[2ex]In the original equation, replace $x$ with its value $7$: \\[2ex]$5 + x = 12$ \\[2ex]$5 + 7 = 12$ \\[2ex]$12 = 12 \quad \text{(The equality is correct.)}$ \end{flushleft}$

$\fontsize{15}{10}\selectfont \begin{flushleft} \raggedright PART2 \textbf{Solution:} \\[2ex]\textbf{Step 1: Define the variable.} \\Let $x$ be the number of apples Emma bought on Tuesday. \\[2ex] \textbf{Step 2: Write the equation.} \\[2ex]$ 5 + x = 12 $ \textit{\textcolor[HTML]{0000FF}{(5 apples that Emma had on Monday, plus $x$ apples she bought on Tuesday equals 12.)}}\\[2ex] % the [HTML]{0000FF} is for the blue color \textbf{Step 3: Solve the equation.} \\[2ex]Since the purpose is to find out the value of $x$, we need to isolate it. To do this, subtract 5 from both sides of the equation. According to the property of equality, "any operation performed on one side of the equation must also be performed on the other side." \\[2ex]$5 - 5 + x = 12 - 5$ \\[2ex]$0 + x = 7$ \\[2ex]$x = 7$ \\[2ex] \textbf{Step 4: Answer.} \\[2ex]\colorbox{yellow}{Emma bought $7$ apples on Tuesday.} \\[2ex] \textbf{Step 5: Verify.} \\[2ex]In the original equation, replace $x$ with its value $7$: \\[2ex]$5 + x = 12$ \\[2ex]$5 + 7 = 12$ \\[2ex]$12 = 12 \quad \text{(The equality is correct.)}$ \end{flushleft}$

Exercise 2

In the following equation, find the value of “x” to make the equation true.

x – 8 = 15

Solution:

Step1: Add 8 to both sides of the equation (before and after the “=” sign.)

x – 8 + 8 = 15 + 8

x + 0 = 23

x = 23

Step2: Answer.

For the equation x – 8 = 15 to be true, the value of “x” must be 23.

Step3: Verify.

In the original equation, just replace “x” by its value 23.

x – 8 = 15

23 – 8 = 15

15 = 15.

Exercise 3

Alex has certain amount of candies. He gives away 4 candies and has 10 left. How many candies did he start with?

Solution:

Step1: Write the equation.

Assume “x” is the number of candies Alex started with.

x – 4 = 10

Alex started with “x” candies, minues 4 that he gives away remaining 10 left.

Step2: Solve the equation.

– Add 4 to each side of the equation

x – 4 = 10.

x – 4 + 4 = 10 + 4. (This can also be written as x + 4 – 4 = 10 + 4.)

x + 0 = 14.

x = 14.

Step3: Answer.

Alex started with 14 candies.

Step4: Verify.

In the original equation, just replace “x” by its value 14.

x – 4 = 10.

14 – 4 = 10.

10 = 10.

Exercise 4

In the following equation, find the value of “x” to make the equation true.

3x = 24.

Solution:

Step1: Applying the property of equality, divide by 3 both sides of the equation (before and after the “=” sign.) Why? Because the variable “x” is being multipled by the number 3 and we need the variable “x” isolated. If instead of 3 we had 4 then, we would divide by 4.

3x = 24.

3x/3 = 24/3.

1.x = 24/3 (Because 3 divided by 3 = 1. And since 1 multiplied by x = x, we don’t need to write the number 1.)

x = 8 (Because 24 divided by 3 = 8.)

Step2: Answer.

For the equation 3x = 24 to be true, the value of “x” must be 8.

Step3: Verify.

In the original equation, just replace “x” by its value 8.

3x = 24.

3*8 = 24. (3x is the same as 3 multiplied by x. In this case, 3 multiplied by 8.)

24 = 24.

Exercise 5

If 5 pencils cost $10 in total, what is the cost of each pencil?

Solution:

Step1: Write the equation.

Assume “x” as the price of each pencil.

5x= 10. (Because 5 pencils multiplied by the price of each pencil = 10.)

Step2: Solve the equation.

Applying the property of equality, divide by 5 both sides of the equation (before and after the “=” sign.) Why? Because the variable “x” is being multipled by the number 5 and we need the variable “x” isolated. If instead of 5 we had 8 then, we would divide by 8.

5x = 10.

5x/5 = 10/5.

1.x = 10/5. (Because 5 divided by 5 = 1. And since 1 multiplied by x = x, we don’t need to write the number 1.)

x = 2 (Because 10 divided by 5 = 2.)

Step3: Answer.

Each pencil costs $2.

Step4: Verify.

In the original equation written on the Step1, just replace “x” by its value 2.

5x = 10.

5*2 = 10 (5x is the same as 5 multiplied by x. In this case, 5 multiplied by 2.)

10 = 10.