Question

1. if the product of 5 and a number is increased by 9 and this sum is multiplied by 3, the result is 11 greater than the opposite of the number. find the number.
2. if the product of 5 and a number is increased by 8, the result is -42. what is the number?
3. if the product of 6 and a number is decreased by 5, the result is -35. what is the number?

Explanation:

Problem 13:

Step1: Define the number

Let the number be \( x \). The opposite of the number is \( -x \).

Step2: Translate the problem into an equation

The product of 5 and the number is \( 5x \). Increased by 9: \( 5x + 9 \). Multiplied by 3: \( 3(5x + 9) \). This result is 11 greater than the opposite of the number: \( 3(5x + 9) = -x + 11 \).

Step3: Solve the equation

Expand the left side: \( 15x + 27 = -x + 11 \).
Add \( x \) to both sides: \( 16x + 27 = 11 \).
Subtract 27 from both sides: \( 16x = 11 - 27 = -16 \).
Divide by 16: \( x = \frac{-16}{16} = -1 \).

Step1: Define the number

Let the number be \( x \).

Step2: Translate the problem into an equation

The product of 5 and the number is \( 5x \). Increased by 8: \( 5x + 8 \). This equals -42: \( 5x + 8 = -42 \).

Step3: Solve the equation

Subtract 8 from both sides: \( 5x = -42 - 8 = -50 \).
Divide by 5: \( x = \frac{-50}{5} = -10 \).

Step1: Define the number

Let the number be \( x \).

Step2: Translate the problem into an equation

The product of 6 and the number is \( 6x \). Decreased by 5: \( 6x - 5 \). This equals -35: \( 6x - 5 = -35 \).

Step3: Solve the equation

Add 5 to both sides: \( 6x = -35 + 5 = -30 \).
Divide by 6: \( x = \frac{-30}{6} = -5 \).

Answer:

The number is \(-1\).

Problem 14: