Problem 12 - Elementary Algebra

Question

In the following exercises, solve using the problem solving strategy for word problems. Remember to write a complete sentence to answer each question. Marc just bought an SUV for \(\$ 54,000\). This is \(\$ 7,400\) less than twice what his wife paid for her car last year. How much did his wife pay for her car?

Step-by-Step Solution

Answer

Marc's wife paid \( \$ 30,700 \) for her car.

Step 1: Understand the Problem

We need to find out how much Marc's wife paid for her car. We know that Marc's SUV cost \( \$ 54,000 \) and this amount is \( \$ 7,400 \) less than twice the amount his wife paid for her car.

Step 2: Define Variables

Let \( x \) be the amount Marc's wife paid for her car.

Step 3: Set Up the Equation

According to the problem, Marc's SUV price \( \$ 54,000 \) is \( \$ 7,400 \) less than twice the amount his wife paid for her car: \[ 54,000 = 2x - 7,400 \]

Step 4: Solve the Equation

Add \( 7,400 \) to both sides of the equation to isolate the term with \( x \): \[ 54,000 + 7,400 = 2x \] Then simplify: \[ 61,400 = 2x \] Divide both sides by \( 2 \) to solve for \( x \): \[ x = \frac{61,400}{2} = 30,700 \]

Step 5: Write the Answer in a Complete Sentence

Marc's wife paid \( \$ 30,700 \) for her car.

Key Concepts

algebraic equations

Algebraic equations are crucial tools in solving a wide array of mathematical problems, including word problems. An algebraic equation shows the relationship between different quantities. For instance, in our problem, the equation we created was based on Marc's SUV price and his wife's car price. Algebraic equations use symbols and numbers to represent real-life situations. Here, we have mathematical expressions on both sides of the equation that need to balance. Equations can be solved by performing operations like addition, subtraction, multiplication, and division.

problem solving

Problem solving in math often involves breaking down the problem into manageable steps. Understanding the problem thoroughly is the first crucial step. In our exercise, we identified that the problem asks for the price Marc's wife paid for her car. Next, we'd set up variables and equations. Defining clear steps and using logical reasoning ensures we don't miss critical details. Finally, we solve the equation and interpret the result, always ensuring it fits the problem's context. Remember, clear and methodical problem solving makes even complex word problems much simpler!

variables definition

A variable in algebra represents an unknown quantity. In the given problem, we needed to find out how much Marc's wife paid for her car, so we let \( x \) symbolize that amount. Variables are placeholders that can take different values and are typically represented by letters such as \( x, y, \) or \( z \). Defining variables correctly is essential because they form the basis for writing our algebraic equations.By assigning a variable to an unknown quantity, we can link it with known quantities and use mathematical operations to find its value.

linear equations

Linear equations are algebraic equations where the highest power of the variable is 1. These are simple yet powerful tools for solving problems. In our exercise, the equation we set up was: \[ 54,000 = 2x - 7,400 \] This is a linear equation because the variable \( x \) is not raised to any power higher than 1. We used basic algebraic techniques to solve it, like adding or subtracting numbers from both sides and dividing both sides by a number. Linear equations usually have a single solution and are widely used in various real-life scenarios.