Solving Perimeter Problems with Linear Equations

Equations, Inequalities, and Problem Solving

Solving Perimeter Problems Using Linear Equations

Perimeter problems often require translating a geometric scenario into an algebraic equation. In this example, the perimeter of a triangle is given, and the lengths of its sides are expressed in terms of a variable. The task is to find the value of the variable and then determine the length of each side.

• Perimeter Definition: The perimeter of a figure is the total distance around it, found by summing the lengths of all sides.

• Translating Words to Equations: Express the relationship in words, then translate to an equation. For a triangle:

  • In words: perimeter = side 1 + side 2 + side 3

  • In equation: $224 = x + 6x + (11x - 10)$

• Combining Like Terms: Remove parentheses and combine terms:

  • $x + 6x + 11x - 10 = 224$

  • $18x - 10 = 224$

• Solving for the Variable: Use algebraic properties to isolate $x$:

  • Add 10 to both sides: $18x - 10 + 10 = 224 + 10$

  • $18x = 234$

  • Divide both sides by 18: $x = \frac{234}{18}$

  • $x = 13$

• Finding Side Lengths: Substitute $x$ back into the expressions for each side:

  • Side 1: $x = 13$

  • Side 2: $6x = 6 \times 13 = 78$

  • Side 3: $11x - 10 = 11 \times 13 - 10 = 133$

• Verification: Check that the sum matches the given perimeter:

  • $13 + 78 + 133 = 224$

Example: If a triangle has sides $x$, $6x$, and $11x - 10$ and a perimeter of 224 feet, solve for $x$ and find the length of each side.

Step-by-step Solution:

• Write the equation: $x + 6x + (11x - 10) = 224$

• Combine like terms: $18x - 10 = 224$

• Add 10: $18x = 234$

• Divide by 18: $x = 13$

• Side 1: $13$ feet

• Side 2: $78$ feet

• Side 3: $133$ feet

Application: This method can be used for any polygon where side lengths are given in terms of a variable and the perimeter is known.