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2. Solving Linear Equations in One Variable
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Today, we will explore linear equations. A linear equation in one variable looks like this: 𝑎𝑥 + 𝑏 = 0. Can anyone tell me what each letter represents?
Is 𝑎 a constant and 𝑥 a variable?
Excellent! Yes, 𝑎 is a constant, whereas 𝑥 is the variable we aim to solve for. Can anyone give me an example of such an equation?
How about 2𝑥 + 3 = 7?
Perfect! Now let's discuss how to solve these types of equations.
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The first step in solving linear equations is simplifying both sides. Who remembers what that means?
It means combining like terms and getting rid of parentheses!
Exactly! Now, what comes next after simplifying?
Move the variable to one side and the constants to the other?
Correct! Isolation of the variable is crucial. Can someone describe how we do that?
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Create a free accountLet's practice solving the equation 3𝑥 - 7 = 11 together. What should our first step be?
Add 7 to both sides to eliminate the constant from the left side.
Great! So what does our equation look like now?
It becomes 3𝑥 = 18.
Exactly! Now, how do we isolate 𝑥?
By dividing both sides by 3, we get 𝑥 = 6.
Let's verify that! What do we do next?
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Create a free accountVerification is critical in our process. Can anyone explain why?
It ensures that our answer actually works in the original equation!
Correct! So, if we substitute 𝑥 = 6 back into our original equation, what will happen?
We will have 3(6) - 7 = 11, which simplifies to 11 = 11!
That's exactly right! Who can summarize what we've learned about solving linear equations?
Overview
Short Summary
This section focuses on the systematic approach to solving linear equations in one variable, including methods and examples.
Medium Summary
Students will learn how to solve linear equations in one variable through a step-by-step process, understanding the importance of simplifying, isolating the variable, and verifying solutions. Several examples will guide students through each stage of the problem-solving process.
Detailed Summary
Solving Linear Equations in One Variable
In this section, we delve into the methods for solving linear equations in one variable, which are crucial for mastering algebra. Linear equations typically come in the form of 𝑎𝑥 + 𝑏 = 0, where 𝑎 and 𝑏 are real numbers and 𝑥 is the variable we need to solve for. The goal is to find the value of 𝑥 that satisfies the equation.
Steps to Solve Linear Equations:
- Simplify both sides: Begin by removing any parentheses or combining like terms to simplify the equation as much as possible.
- Move variables to one side and constants to the other: Rearrange the equation so that all terms containing the variable are on one side and constant terms on the other.
- Isolate the variable: Use inverse operations to get the variable alone on one side of the equation.
- Check the solution: Substitute your solution back into the original equation to ensure it satisfies the equation.
Example Walkthrough:
For instance, to solve the equation 3𝑥−7 = 11:
- Step 1: Add 7 to both sides to get 3𝑥 = 18.
- Step 2: Divide both sides by 3, resulting in 𝑥 = 6.
- Final Verification: Substituting back gives us 3(6)−7 = 11, confirming our solution.
By practicing these steps, students will build confidence in solving linear equations, vital for tackling more complex algebraic expressions and real-world applications.
Audio Book
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Create a free account🔸 Steps to Solve
- Simplify both sides: Remove parentheses, combine like terms.
- Move variables to one side and constants to the other.
- Isolate the variable using inverse operations.
- Check the solution by substituting it back into the original equation.
Detailed Explanation
To solve a linear equation, we need to follow systematic steps:
- Simplify: Begin by simplifying both sides of the equation. This involves removing any parentheses and combining like terms, making it easier to work with.
- Move Terms: Next, we'll rearrange the equation to get all variable terms on one side and constant terms on the other side. This often involves adding or subtracting terms from both sides.
- Isolate the Variable: Our goal is to get the variable by itself. We achieve this by using inverse operations like addition, subtraction, multiplication, or division.
- Check the Solution: Finally, we verify our answer by substituting it back into the original equation to ensure both sides are equal. This step helps confirm that we've solved the equation correctly.
Examples & Analogies
Imagine you're trying to balance a scale. On one side, you have a box with an unknown weight, and on the other, you have weights that total to a known amount. To figure out the weight of the box, you need to take away known weights (simplifying) and keep adjusting until the two sides balance (isolation). Finally, you check the box's weight against the known weights to see if it matches.
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Create a free account🔹 Examples Example 1: Solve 3𝑥−7 = 11 Solution: Add 7 to both sides: 3𝑥 = 18 Divide by 3: 𝑥 = 6
Detailed Explanation
Let's break down the example of solving the equation 3𝑥 - 7 = 11 step by step:
- Start with the equation: 3𝑥 - 7 = 11.
- Simplify: The first operation we perform is adding 7 to both sides to eliminate the '-7' from the left side. So we have:
- 3𝑥 - 7 + 7 = 11 + 7, which simplifies to 3𝑥 = 18.
- Isolate the variable: Next, we divide both sides of the equation by 3 to solve for 𝑥:
- 3𝑥 / 3 = 18 / 3, resulting in 𝑥 = 6.
- Check the solution: To ensure this solution is correct, we substitute 6 back into the original equation and verify:
- 3(6) - 7 = 11, which simplifies to 18 - 7 = 11, confirming our solution is accurate.
Examples & Analogies
Think of a linear equation like planning money for a pizza party with friends. Suppose the total cost of the pizza is 11. So the equation becomes 3x = 11. To find out each person’s share, you simply solve the equation by dividing both sides by 3, which gives x = 11 ÷ 3 ≈ 3.67. This means each friend pays about $3.67.
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Key Concepts
Core takeaways and short definitions to help you quickly recall the key ideas from this section.
Simplifying Equations: The process of combining like terms to reduce the equation to its simplest form before solving.
Isolating the Variable: The technique of rearranging the equation to get the variable on its own side.
Verification: The practice of substituting the solution back into the original equation to confirm it is correct.
Examples
Memory Aids
Interactive tools to help you remember key concepts