Problem 87 - Introductory Algebra for College Students

Question

In Massachusetts, speeding fines are determined by the formula $$F=10(x-65)+50$$ where \(F\) is the cost, in dollars, of the fine if a person is caught driving \(x\) miles per hour. Use this formula to solve. If a fine comes to \(\$ 250,\) how fast was that person driving?

Step-by-Step Solution

Answer

The person was driving at 85 miles per hour.

Step 1: Substitution

Substitute the fine $F$ in the given formula with $250: \[ 250 = 10(x - 65) + 50 \]

Step 2: Reorganize the equation

Now, we can simplify the equation by subtracting 50 from both sides: \[ 200 = 10(x - 65) \]

Step 3: Solve for x

To find the speed, divide both sides by 10: \[ 20 = x - 65 \] Then, add 65 to both sides to solve for \( x \): \[ x = 20 + 65 \]

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y), and operators (such as +, −, ×, ÷). The purpose of an algebraic expression is to represent a specific value or relationship in a compact form. For instance, in the exercise, \[ F=10(x-65)+50 \] is an algebraic expression that represents the fine charged for speeding in Massachusetts.

Algebraic expressions are essential because they allow us to describe real-world situations mathematically, which then enables us to solve problems. The equation here uses variables to stand for the cost of the fine (F) and the speed of the driver (x). By manipulating this expression, we can solve for unknown quantities, such as determining the speed that resulted in a $250 fine.

Linear Equation Problem Solving

Linear equation problem solving involves finding the value of the variable that makes the equation true. These equations are called 'linear' because their graphs are straight lines. The general form of a linear equation in one variable x is ax + b = c, where a, b, and c are constants and x is the variable we're solving for.

When solving such equations, our objective is to isolate the variable on one side of the equation. This process often involves several steps, such as redistributing terms, combining like terms, and finally reversing operations used on the variable. In our problem, by following a calculated step-by-step approach, we have determined the driver's speed by isolating the variable x and calculating its value.

Equation Simplification

Equation simplification refers to the process of transforming a complex equation into a simpler or more standard form without changing its solutions. The goal is to make the equation easier to understand and solve. To simplify an equation, we perform operations on both sides of the equation that will move us towards having the variable alone on one side.

For example, in our textbook problem, we started with 250 being equal to 10 times the quantity of (x minus 65) plus 50. The steps to simplify include subtracting 50 from both sides to eliminate the constant on one side, and then dividing by 10 to reverse the multiplication of the variable. These operations reduce the equation to x equals a number, which is a much simpler form and readily tells us the speed associated with the fine.