Question

1. if the sum of twice a number and -12 is multiplied by 2, the result if 11 greater than the opposite of the number. find the number.
2. find a number which decreased by 42 equals 5 times its opposite.
3. find a number which decreased by 21 equals twice the opposite of the number.
4. the sum of 3 times a number and 70 equaled -50. find the number.
5. if the product of 3 and a number is increased by 7, the result is 23 greater than the number. find the number.
6. twice the sum of 3 times a number and 60 is 155 greater than the opposite of the number. find the number.

Explanation:

Problem 7

Step 1: Define the variable

Let the number be \( x \).

Step 2: Translate the problem into an equation

The sum of twice a number (\( 2x \)) and -12 is \( 2x - 12 \). Multiply this by 2: \( 2(2x - 12) \). The opposite of the number is \( -x \). The result is 11 greater than the opposite of the number, so:

$$ 2(2x - 12) = -x + 11 $$

Step 3: Solve the equation

Expand the left side: \( 4x - 24 = -x + 11 \)
Add \( x \) to both sides: \( 5x - 24 = 11 \)
Add 24 to both sides: \( 5x = 35 \)
Divide by 5: \( x = 7 \)

Step 1: Define the variable

Let the number be \( x \).

Step 2: Translate the problem into an equation

A number decreased by 42 is \( x - 42 \). Its opposite is \( -x \), and 5 times its opposite is \( -5x \). So:

$$ x - 42 = -5x $$

Step 3: Solve the equation

Add \( 5x \) to both sides: \( 6x - 42 = 0 \)
Add 42 to both sides: \( 6x = 42 \)
Divide by 6: \( x = 7 \)

Step 1: Define the variable

Let the number be \( x \).

Step 2: Translate the problem into an equation

A number decreased by 21 is \( x - 21 \). Twice the opposite of the number is \( -2x \). So:

$$ x - 21 = -2x $$

Step 3: Solve the equation

Add \( 2x \) to both sides: \( 3x - 21 = 0 \)
Add 21 to both sides: \( 3x = 21 \)
Divide by 3: \( x = 7 \)

Answer:

7

Problem 8