Problem 7 - Algebra A Combined Function

Question

Substitute the given values into each given formula and solve for the unknown variable. $$ \begin{aligned} &P=a+b+c ; \quad P=30, a=8, b=10\\\ &\text { of a triangle) } \end{aligned} $$

Step-by-Step Solution

Answer

The unknown variable \( c \) is 12.

Step 1: Identify the Formula

Identify the formula provided which is for the perimeter of a triangle: \( P = a + b + c \). We are given that \( P = 30 \), \( a = 8 \), and \( b = 10 \), and we need to solve for \( c \).

Step 2: Substitute Known Values into the Formula

Substitute the given values into the formula: \( 30 = 8 + 10 + c \).

Step 3: Simplify the Equation

Simplify the equation we obtained in Step 2: \( 30 = 18 + c \).

Step 4: Solve for the Unknown Variable

To find \( c \), subtract 18 from both sides of the equation: \( c = 30 - 18 \).

Step 5: Calculate the Result

Calculate the value of \( c \): \( c = 12 \). This is the length of the missing side of the triangle.

Key Concepts

Perimeter of a Triangle

The perimeter of a triangle is the total length around the shape. Imagine walking along the edges of the triangle; the distance you cover is the perimeter.
The formula to calculate the perimeter, represented by \( P \), is given as \( P = a + b + c \), where \( a \), \( b \), and \( c \) are the lengths of the triangle's sides. For example, if one side measures 8 units, another is 10 units, and you want to find the third side with a perimeter of 30 units, this formula helps you figure it out.
Understanding the concept of perimeter helps in various real-life scenarios, such as when you need to determine how much fencing is required to border a triangular garden.

• Perimeter involves addition of all side lengths.
• Used in everyday tasks like home improvement and craft projects.

Solving Equations

Solving equations means finding the value of the unknown variable that makes the equation true. An equation is like a balance scale; both sides must be equal.
In the context of our problem, to solve for the unknown, you first rewrite the equation using known values. Here, the equation \( 30 = 8 + 10 + c \) involves simplifying one side to isolate the unknown.

• Ensure you perform the same operation on both sides of the equation to keep it balanced.
• Simplify by combining like terms and reducing the equation step by step.

This logical process helps you systematically determine the unknown value, ensuring accuracy and consistency.

Substitution Method

The substitution method is a technique that involves replacing one variable with another value or expression to solve an equation.
Let's say you have an equation with known values for some variables. By substituting these values into the equation, it simplifies your task to solve for the unknown.
In the triangle perimeter example, substituting \( a = 8 \) and \( b = 10 \) directly into the equation \( P = a + b + c \) leads to \( 30 = 8 + 10 + c \), helping you solve for \( c \).

• Replace known variables with their values.
• Simplifies complex equations into simpler form.

This method is effective in algebraic problem-solving, especially when dealing with multiple variables.

Unknown Variable

An unknown variable is a symbol in an equation representing a value that needs to be found. It is often denoted by letters like \( c \), \( x \), or \( y \).
In math problems, identifying the unknown variable is crucial, as it indicates what you're solving for. In this exercise, the unknown was \( c \), one of the sides of the triangle.
To find the value of \( c \), the steps included isolating it on one side of the equation. By subtracting 18 from 30 in the equation \( 30 = 18 + c \), \( c \)'s value was discovered to be 12.

• Identify what information is missing in your problem.
• Use algebraic manipulation to reveal the unknown's value.

Understanding unknowns makes tackling equations smooth and manageable.