Explore

Solving Problems with Formulas Solutions

When solving problems with formulas, start by reading the problem carefully.  
- Name the formula that contains both the value that you are solving for and the values that you already know.
Once you have written down a formula and defined the variables in the problem: 
- Substitute the values that you already know into the formula.  
- Solve for the remaining variable in the formula. 
Remember to label your final answer with the correct units . If no units of measurement are specified, simply use “units.”

Example 1

Determine the formula needed to solve.

Triangle: $P = a + b + c$ Rectangle: $P = 2 l + 2 w$

The perimeter of a rectangle is 48 units. The length is twice the width plus 3. What are the dimensions?

Implement

$P = 2 l + 2 w P = 48 l = 2 w + 3$

$48 = 2 (2 w + 3) + 2 w$

$\begin{array}{rcl} 48 & = & 4 w + 6 + 2 w \\ 48 & = & 6 w + 6 \\ - 6 & - 6 \\ 42 & = & 6 w \\ w & = & 7 \end{array}$

Check
$48 = 2 (17) + 2 (7) 48 = 34 + 14 ✓$

$l = 2 (7) + 3 = 17$

Explain

Name the formula 
Identify the key information
Determine the equation for length using mathematical vocabulary
Substitute values into the formula
Solve for the variable, w
Substitute to find the length, l

The dimensions of the rectangle are 17 units by 7 units.

Example 2

Determine the formula needed to solve.

Triangle: $P = a + b + c$ Rectangle: $P = 2 l + 2 w$

The perimeter of a triangle is 30 inches. Side b is twice the length of side a, and side c is three times the length of side a. Find the dimensions of the triangle.

Implement

$P = a + b + c P = 30, b = 2 a, c = 3 a$

$\begin{array}{rcl} 30 & = & a + (2 a) + (3 a) \\ 30 & = & 6 a \\ a & = & 5 \end{array} b = 2 (5) = 10 c = 3 (5) = 15 Check \begin{array}{rcl} 30 & = & 5 + 10 + 15 \\ 30 & = & 15 + 15 ✓ \end{array}$

Explain

Name the formula
Identify the key information
Determine the equations for b and c
Solve for a, b, and c

The dimensions of the triangle are 5 inches, 10 inches, and 15 inches.