Question

paula sold 45 packages of wrapping paper for the christmas fundraiser. this is three less than twice the number of packages that she sold last year. how many packages did paula sell last year? let____=______ equation:______ the area of a triangle increased by 10 is the same as 30 decreased by 9 times the area. what is the area of the triangle? let__=______ equation:________

Explanation:

First problem:

Step1: Define the variable

Let \( x \) = the number of packages Paula sold last year.

Step2: Form the equation

Since 45 is three less than twice the number of packages sold last year, the equation is \( 2x - 3=45 \).

Step3: Solve the equation

Add 3 to both sides: \( 2x-3 + 3=45 + 3 \), which simplifies to \( 2x=48 \).
Divide both sides by 2: \( x=\frac{48}{2}=24 \).

Second problem:

Step1: Define the variable

Let \( A \) = the area of the triangle.

Step2: Form the equation

The area of a triangle increased by 10 is \( A + 10 \), and 30 decreased by 9 times the area is \( 30-9A \). So the equation is \( A + 10=30 - 9A \).

Step3: Solve the equation

Add \( 9A \) to both sides: \( A+9A + 10=30-9A + 9A \), which gives \( 10A+10 = 30 \).
Subtract 10 from both sides: \( 10A+10 - 10=30 - 10 \), so \( 10A=20 \).
Divide both sides by 10: \( A=\frac{20}{10}=2 \).

Answer:

First problem:
Let \( x \) = the number of packages Paula sold last year. Equation: \( 2x - 3 = 45 \). Paula sold 24 packages last year.
Second problem:
Let \( A \) = the area of the triangle. Equation: \( A + 10=30 - 9A \). The area of the triangle is 2.