Question

practice reasoning about multi - step equations

• study how the example shows using reasoning to find the value of an unknown in an equation. then solve problems 1 - 6.

example
the product of 4 and (3 + x) is equal to 36. what is the value of x?
the equation 4(3 + x) = 36 models this statement.
think: 4 times what number is 36?
since 4·9 equals 36, that means (3 + x) equals 9.
since 3 + x = 9, and 3 + 6 = 9, that means x equals 6.

1. the sum of twice a number, n, and 14 is 30. write an equation that models this statement. then explain how you might use reasoning to find the value of n.
2. is the value of (y + 6) greater in 4(y + 6) = 48 or 12(y + 6) = 48? how do you know?
3. ana wants to place 11 plants on 3 shelves. each shelf holds the same number of plants, p. there are 2 plants that do not fit. model this situation with an equation and a hanger diagram.

Explanation:

Problem 1

Step1: Translate the statement into an equation.

"Twice a number \( n \)" is \( 2n \), and "the sum of twice a number \( n \) and 14 is 30" gives the equation \( 2n + 14 = 30 \).

Step2: Reason to find \( n \).

Think: What number plus 14 equals 30? Since \( 16 + 14 = 30 \), that means \( 2n = 16 \).

Step3: Solve for \( n \).

What number times 2 equals 16? Since \( 2 \times 8 = 16 \), \( n = 8 \).

Step1: Analyze \( 4(y + 6) = 48 \).

Think: 4 times what number is 48? Since \( 4 \times 12 = 48 \), \( y + 6 = 12 \).

Step2: Analyze \( 12(y + 6) = 48 \).

Think: 12 times what number is 48? Since \( 12 \times 4 = 48 \), \( y + 6 = 4 \).

Step3: Compare the values of \( y + 6 \).

Since \( 12 > 4 \), the value of \( (y + 6) \) is greater in \( 4(y + 6) = 48 \).

Step1: Translate the situation into an equation.

There are 3 shelves with \( p \) plants each, so \( 3p \), and 2 plants that do not fit, and the total number of plants is 11. So the equation is \( 3p + 2 = 11 \).

Step2: Draw the hanger diagram (description).

• One side of the hanger has three equal parts labeled \( p \) and one part labeled 2.
• The other side of the hanger has a part labeled 11, showing they are balanced.

Answer:

The equation is \( 2n + 14 = 30 \), and the value of \( n \) is 8.

Problem 2