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Questions and Answers
What is the standard form of a linear equation in one variable?
What is the standard form of a linear equation in one variable?
To isolate the variable in a linear equation, you can only use multiplication.
To isolate the variable in a linear equation, you can only use multiplication.
False
What should you do first when you approach a word problem to set up a linear equation?
What should you do first when you approach a word problem to set up a linear equation?
Identify the variable and determine what it represents.
For the equation $3x - 7 = 2$, the first step to solve for $x$ is to _____ both sides by 1.
For the equation $3x - 7 = 2$, the first step to solve for $x$ is to _____ both sides by 1.
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Match the linear equation with its corresponding solution:
Match the linear equation with its corresponding solution:
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When checking the solution of an equation, what method is typically used?
When checking the solution of an equation, what method is typically used?
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In the equation $2x - 4 = 10$, the solution is found by first adding 4 to both sides.
In the equation $2x - 4 = 10$, the solution is found by first adding 4 to both sides.
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In the problem 'Twice a number decreased by four equals ten', what is the equation formed?
In the problem 'Twice a number decreased by four equals ten', what is the equation formed?
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To solve the equation $3y + 5 = 20$, you first need to _____ both sides by 3.
To solve the equation $3y + 5 = 20$, you first need to _____ both sides by 3.
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What is the final solution for the equation $x + 5 = 12$?
What is the final solution for the equation $x + 5 = 12$?
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Study Notes
Linear Equations in One Variable
Solving Techniques
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Definition: An equation that can be expressed in the form ( ax + b = 0 ), where ( a ) and ( b ) are constants and ( x ) is the variable.
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Isolating the Variable:
- Addition/Subtraction: Add or subtract terms from both sides to isolate ( x ).
- Multiplication/Division: Multiply or divide both sides by a non-zero constant to solve for ( x ).
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Standard Steps:
- Simplify both sides of the equation.
- Move all terms involving ( x ) to one side and constant terms to the other side.
- Combine like terms.
- Isolate ( x ) using basic algebraic operations.
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Examples:
- ( 2x + 5 = 11 ) → ( 2x = 6 ) → ( x = 3 )
- ( 3x - 7 = 2 ) → ( 3x = 9 ) → ( x = 3 )
Word Problems
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Identifying Variables: Determine what the variable represents based on the problem context.
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Setting Up Equations: Translate the word problem into a linear equation.
- Identify relationships and values described in the problem.
- Use operations to express these relationships mathematically.
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Examples:
- "Twice a number decreased by four equals ten."
- Let ( x ) be the number: ( 2x - 4 = 10 )
- "Three times a number plus five is equal to twenty."
- Let ( y ) be the number: ( 3y + 5 = 20 )
- "Twice a number decreased by four equals ten."
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Solving the Equation: Follow standard solving techniques outlined above after forming the equation.
Checking Solutions
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Substitution: Substitute the found solution back into the original equation to verify correctness.
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Steps:
- Take the solution obtained (e.g., ( x = 3 )).
- Replace ( x ) in the original equation.
- Simplify and see if both sides are equal.
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Examples:
- For ( x = 3 ) in ( 2x + 5 = 11 ):
- Check: ( 2(3) + 5 = 6 + 5 = 11 ) (True)
- For ( y = 5 ) in ( 3y + 5 = 20 ):
- Check: ( 3(5) + 5 = 15 + 5 = 20 ) (True)
- For ( x = 3 ) in ( 2x + 5 = 11 ):
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Conclusion: If both sides are equal after substitution, the solution is correct. If not, re-evaluate the steps taken to find the solution.
Linear Equations in One Variable
Solving Techniques
- Linear equations take the form ( ax + b = 0 ), where ( a ) and ( b ) are constants, and ( x ) is the variable.
- Isolate ( x ) using two main strategies:
- Addition/Subtraction: Adjust both sides of the equation by adding or subtracting constants.
- Multiplication/Division: Scale both sides with non-zero constants to derive the value of ( x ).
- Follow these standard steps to solve:
- Simplify expressions on both sides of the equation.
- Shift all ( x )-related terms to one side, moving constant terms across the equality.
- Combine like terms for clarity and ease.
- Use algebraic operations to isolate ( x ).
- Example of solving:
- From ( 2x + 5 = 11 ) to ( 2x = 6 ), resulting in ( x = 3 ).
- Transforming ( 3x - 7 = 2 ) to find ( x = 3 ) after ( 3x = 9 ).
Word Problems
- Identify what the variable symbolizes within the problem's context.
- Translate the situation into a mathematical equation by:
- Recognizing relationships and quantities described in the text.
- Utilizing appropriate operations to formulate these relationships quantitatively.
- Example translations:
- For “Twice a number decreased by four equals ten,” let ( x ) be the number translates to ( 2x - 4 = 10 ).
- The phrase “Three times a number plus five is equal to twenty” translates to ( 3y + 5 = 20 ) when letting ( y ) be the unknown.
Checking Solutions
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To verify solutions, substitute the calculated value back into the original equation.
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Steps for verification:
- Utilize the solution obtained (e.g., ( x = 3 )).
- Insert this value into the original equation and simplify to check if both sides of the equation are equal.
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Example checks:
- Substituting ( x = 3 ) into ( 2x + 5 = 11 ) confirms correctness as ( 2(3) + 5 = 11 ).
- For ( y = 5 ) in the equation ( 3y + 5 = 20 ), checking gives ( 3(5) + 5 = 20 ), affirming it is true.
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Conclusion drawn from checks indicates that if equality holds post-substitution, the solution is valid, otherwise re-evaluation of steps is necessary.
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Description
Test your understanding of linear equations in one variable! This quiz covers solving techniques, isolating the variable, and applying methods to word problems. Perfect for students looking to strengthen their algebra skills.
