Goals
1. Understand and solve first-degree polynomial equations of the form ax + b = c using the properties of equality.
2. Apply knowledge of first-degree equations to real-life situations, such as calculating taxi fares.
Contextualization
First-degree equations are part of our everyday lives. For instance, when you're planning a taxi ride, you’ll want to figure out the total fare, which consists of a base fare and a per-mile charge. Being able to solve first-degree equations helps you estimate the cost of your trip accurately and efficiently. For example, if the base fare for a taxi ride is $6.00 and the charge per mile is $1.80, you can easily determine the total cost for various distances.
Subject Relevance
To Remember!
Definition of a First-Degree Equation
A first-degree equation is a mathematical expression that can be arranged in the form ax + b = c, where 'a', 'b', and 'c' are constants and 'x' is the variable. The key characteristic of these equations is that the variable appears only to the first power.
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First-degree equations are linear.
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The solution is the value of 'x' that satisfies the equation.
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These equations are essential for solving problems related to proportions and costs.
Properties of Equality
The properties of equality are mathematical rules that let you manipulate equations to isolate the variable. The main properties include: addition, subtraction, multiplication, and division by the same number on both sides of the equation.
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Addition: If a = b, then a + c = b + c.
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Subtraction: If a = b, then a - c = b - c.
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Multiplication: If a = b, then a * c = b * c, provided c ≠ 0.
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Division: If a = b, then a / c = b / c, provided c ≠ 0.
Solving First-Degree Equations in the Form ax + b = c
To solve a first-degree equation, the goal is to isolate the variable 'x' on one side of the equation. This generally requires using the properties of equality to simplify the expression.
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Isolating 'x' might take more than one step.
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Start by removing constants from the side of the variable.
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Then, divide or multiply as necessary to solve the equation.
Practical Applications
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Calculating taxi fares, incorporating a base charge and a variable charge per mile.
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Managing personal finances, like drafting a budget that includes fluctuating income and expenses.
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Evaluating service contracts, such as phone and internet plans, which involve fixed and variable rates.
Key Terms
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First-Degree Equation: A linear equation in the form ax + b = c.
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Properties of Equality: Mathematical rules for manipulating equations.
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Fixed Rate: A cost that remains unchanged regardless of usage.
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Variable Rate: A cost that changes based on usage amounts.
Questions for Reflections
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How can first-degree equations assist us in everyday tasks, such as planning a journey?
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What challenges might arise when solving first-degree equations, and how could you address them?
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In what ways can understanding the properties of equality enhance the efficiency of solving mathematical problems?
Planning a Taxi Trip
Utilize your understanding of first-degree equations to estimate the cost of a taxi ride over varying distances.
Instructions
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Consider that a taxi ride has a base fare of $6.00 and a charge of $1.80 per mile.
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Formulate the equation that represents the total cost (c) for a trip of x miles.
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Solve the equation to find the total cost for journeys of 5 miles, 10 miles, and 20 miles.
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Evaluate the costs and reflect on how minor adjustments in travel distance can impact the total fare.
