Problem 48 - Algebra for College Students

Question

Solve each of Problems \(47-62\) by setting up. The perimeter of a triangle is 42 inches. The second side is 1 inch more than twice the first side, and the third side is 1 inch less than three times the first side. Find the lengths of the three sides of the triangle.

Step-by-Step Solution

Answer

The sides of the triangle are 7, 15, and 20 inches.

Step 1: Assign Variables

Let the first side of the triangle be \(x\) inches. Therefore, the second side, which is 1 inch more than twice the first side, can be expressed as \(2x + 1\). The third side, which is 1 inch less than three times the first side, can be written as \(3x - 1\).

Step 2: Formulate the Equation

The perimeter of the triangle is the sum of its sides, which we know equals 42 inches. Set up the equation representing the perimeter: \[ x + (2x + 1) + (3x - 1) = 42 \]

Step 3: Simplify the Equation

Combine like terms in the equation.\[ x + 2x + 1 + 3x - 1 = 42 \]Simplify to:\[ 6x = 42 \]

Step 4: Solve for the First Side

To find \(x\), divide both sides of the equation by 6:\[ x = \frac{42}{6} \]Thus, \(x = 7\). So, the first side is 7 inches.

Step 5: Calculate the Other Sides

Use the value of \(x\) to find the lengths of the other sides. The second side is:\[ 2x + 1 = 2(7) + 1 = 15 \] inches.The third side is:\[ 3x - 1 = 3(7) - 1 = 20 \] inches.

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra that involves finding the value of unknown variables. In this exercise, we start by using algebra to form an equation based on the description of a triangle's sides and its perimeter. This requires setting up an equation that equates the expressions for the sides of the triangle to the given perimeter.

• Assign variables to represent unknown quantities. For example, the first side is assigned as \( x \).
• Express other unknowns in terms of this variable. The second and third sides are expressed as \( 2x + 1 \) and \( 3x - 1 \).

Once the equation is set up, combine like terms. In the formed equation \[ x + (2x + 1) + (3x - 1) = 42 \]the terms simplify to \[ 6x = 42 \]. Here, the key step is isolating \( x \) to find its value. Divide both sides by the coefficient of \( x \) (which is 6) to find \( x = 7 \). Solving the equation involves selecting appropriate operations to isolate the variable. This could mean adding, subtracting, or dividing terms to simplify and solve for the desired expression.

Perimeter of Triangle

The perimeter of a triangle is the total length around the triangle, calculated by adding up the lengths of its three sides. Understanding how to find the perimeter is crucial when setting up problems like this one, where specific relationships between sides are given.In our problem, the triangle’s perimeter is 42 inches. We use this information to establish an equation relating all three sides:

• First side \( = x \)
• Second side \( = 2x + 1 \)
• Third side \( = 3x - 1 \)

The equation for the perimeter of our triangle is then \[ x + (2x + 1) + (3x - 1) = 42 \]. This equation uses the fundamental idea that the perimeter is the sum of all side lengths. Solving this will help find the exact lengths of the sides.Remember, the key idea in perimeter problems is the sum of all sides should match the given perimeter. By correctly solving this equation, you'll discover each side's length matches the conditions provided in the problem.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. In this problem, expressions are used to model the relationships between the sides of a triangle.We represent the sides of the triangle with algebraic expressions based on a single variable \( x \):

• The first side is simply \( x \).
• The second side is \( 2x + 1 \), indicating it is 1 more than twice the first side.
• The third side is \( 3x - 1 \), signifying it is 1 less than three times the first side.

These expressions tell us how each side relates to \( x \) and, ultimately, how they relate to each other. Simplifying an expression means reducing it to its simplest form, which was done when combining terms of the perimeter equation. Expressions are not only used for setting up equations but also for simplifying the math involved in solving them.In algebra, working with expressions effectively often involves recognizing patterns and making sure they reflect the problem's conditions accurately, using operations like addition, subtraction, multiplication, and division.