Chapter 3: Solving Linear Equations
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Understanding Linear Equations
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Today, we're diving into what linear equations are. A linear equation is a mathematical statement that shows two expressions are equal. Can someone remind me what we mean by an expression?
An expression is a combination of numbers, variables, and operations but does not have an equals sign.
Exactly! Now, what do we call the unknown quantity in a linear equation?
That would be the variable.
Correct! The variable is often represented by letters like x or y. Letβs take a step deeper. What does it mean to solve a linear equation?
To solve it means to find the value of the variable that makes the equation true.
Great! Remember, we use inverse operations to isolate the variable. Can anyone give an example of inverse operations?
Adding and subtracting are inverse operations!
Wonderful! To summarize, linear equations balance two expressions and require the use of inverse operations to solve for the variable.
Solving One-Step Equations
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Letβs solve our first one-step equation together! What do we do to solve `x + 5 = 12`?
We should subtract 5 from both sides.
Exactly! So, what do we get after that?
We get `x = 7`.
Perfect! Letβs try another. Solve `4y = 20`. Whatβs the first step?
We need to divide both sides by 4.
Great! Whatβs our solution?
y equals 5!
Exactly! Remember, each operation we perform has to be mirrored on both sides to maintain balance.
Two-Step Linear Equations
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Now, let's up the ante with two-step equations. Who can help me solve `2x + 3 = 11`?
First, we need to subtract 3 from both sides.
Correct! What do we have now?
2x = 8.
Excellent! Whatβs the next step?
Now we divide both sides by 2.
What do we get after that?
x equals 4!
Spot on! Remember, with two-step equations, we always undo addition/subtraction first, then tackle multiplication/division.
Equations with Variables on Both Sides
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Letβs tackle a more challenging problem: `3x + 4 = x + 12`. What's the first step?
We can start by keeping all the x terms on one side, so let's subtract x from both sides.
Good thinking! What does that give us?
2x + 4 = 12.
Excellent! Now how do we isolate x?
Subtract 4 from both sides to get `2x = 8`.
Yes! And whatβs our final step to solve for x?
Divide both sides by 2, so x equals 4!
Outstanding! Remember, when handling variables on both sides, we always try to gather them on one side of the equation.
Complex Multi-Step Equations
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Finally, letβs solve multi-step equations like `4(x - 2) = 12`. Who can guide us through solving this?
We first need to expand the brackets.
Correct! What does that give us?
It becomes `4x - 8 = 12`.
Great! Whatβs our next move?
We add 8 to both sides to isolate the term with x.
And what does our equation look like now?
Now we have `4x = 20`.
Perfect! How do we finish solving for x?
We divide by 4, which gives us x = 5!
Exactly! See how expanding, isolating, and finally solving are crucial steps in multi-step equations?
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
In this section, students will learn how to solve various types of linear equations through step-by-step processes. Key concepts include understanding inverse operations, handling equations with multiple steps, and recognizing how to isolate variables effectively. The relevance of these skills in applying mathematical reasoning to real-world scenarios is emphasized.
Detailed
Detailed Summary
This section focuses on solving linear equations, which are fundamental to algebra and essential for understanding mathematical relationships. A linear equation is defined as a mathematical statement that equates two expressions and features the unknown variable raised to the first power.
Key Concepts Covered:
- Definition of an Equation: An equation represents a balance between two expressions connected by an equals sign. The objective of solving equations is to determine the values of the variable(s) that satisfy this equality.
- Inverse Operations: To solve an equation, students must understand inverse operations, ensuring whatever action is performed on one side of the equation is also done on the other side. For example, addition and subtraction are inverse operations, as are multiplication and division.
-
One-Step Linear Equations: These equations involve a single operation (either addition or subtraction, or multiplication or division). For instance, to solve
x + 7 = 15, one would subtract 7 from both sides, yieldingx = 8. -
Two-Step Linear Equations: These include two operations. The general approach is to first eliminate the addition or subtraction to isolate the term with the variable, and then address multiplication or division. An example is
2x + 3 = 11, solved by first subtracting 3 and then dividing by 2 to findx = 4. -
Multi-Step Equations: Involving brackets and like terms, these require expansion and simplification before isolating the variable. An example is
4(x - 2) = 12, which would be solved by expanding to4x - 8 = 12, then isolating the variable. -
Equations with Variables on Both Sides: This section discusses strategies for gathering like terms to isolate the variable on one side, ensuring a clear pathway to the solution. An example of this concept is solving
6x + 5 = 2x + 13, where the strategy involves moving variable terms to one side and constant terms to the other.
This section underscores the relevance of solving linear equations in real-world applications, such as calculating costs, understanding balances, or predicting changes.
Key Concepts
-
Definition of an Equation: An equation represents a balance between two expressions connected by an equals sign. The objective of solving equations is to determine the values of the variable(s) that satisfy this equality.
-
Inverse Operations: To solve an equation, students must understand inverse operations, ensuring whatever action is performed on one side of the equation is also done on the other side. For example, addition and subtraction are inverse operations, as are multiplication and division.
-
One-Step Linear Equations: These equations involve a single operation (either addition or subtraction, or multiplication or division). For instance, to solve
x + 7 = 15, one would subtract 7 from both sides, yieldingx = 8. -
Two-Step Linear Equations: These include two operations. The general approach is to first eliminate the addition or subtraction to isolate the term with the variable, and then address multiplication or division. An example is
2x + 3 = 11, solved by first subtracting 3 and then dividing by 2 to findx = 4. -
Multi-Step Equations: Involving brackets and like terms, these require expansion and simplification before isolating the variable. An example is
4(x - 2) = 12, which would be solved by expanding to4x - 8 = 12, then isolating the variable. -
Equations with Variables on Both Sides: This section discusses strategies for gathering like terms to isolate the variable on one side, ensuring a clear pathway to the solution. An example of this concept is solving
6x + 5 = 2x + 13, where the strategy involves moving variable terms to one side and constant terms to the other. -
This section underscores the relevance of solving linear equations in real-world applications, such as calculating costs, understanding balances, or predicting changes.
Examples & Applications
Example of a one-step equation: Solve for x in x + 5 = 12.
Example of a two-step equation: Solve for x in 2x + 3 = 11.
Example of a multi-step equation: Expand and solve 4(x - 2) = 12.
Example of a variable on both sides: Solve 5v + 2 = 3v + 10.
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
To balance the scale, don't be a fool, do to both sides the same, that's the rule.
Stories
Imagine a seesaw. If one child adds a block, the other child must do the same to keep the seesaw balanced. This is just like keeping equations balanced.
Memory Tools
For solving equations, think S.I.M.P.L.E: S - simplify, I - isolate, M - maintain balance, P - perform operations, L - lessen terms, E - evaluate.
Acronyms
E.Q.U.A.T.E
Equalize both sides
Question the operation
Undo with inverse
Apply operations
Test your solution
Evaluate.
Flash Cards
Glossary
- Equation
A mathematical statement that asserts the equality of two expressions.
- Solution
The value(s) of the variable that make the equation true.
- Inverse Operation
An operation that undoes another operation.
- OneStep Equation
An equation that requires only one operation to solve.
- TwoStep Equation
An equation that involves two operations to find the solution.
- MultiStep Equation
An equation that requires multiple steps and operations, often involving brackets.
- Variable
A symbol used to represent an unknown value, often denoted by letters like x or y.
Reference links
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