Questions and Answers
What are the steps for solving $x$ when $x$ is in the numerator?
- Divide the numbers 2. Multiply all terms by the same value to eliminate fraction denominators 3. Simplify 4. Write $x = $
Solve $\frac{14}{7} = \frac{x}{12}$
$x = 24$
What are the steps for solving $x$ when $x$ is in the denominator?
- Divide the numbers 2. Multiply all terms by the same value to eliminate fraction denominators 3. Simplify 4. Multiply all terms by the same value to eliminate fraction denominators 5. Simplify 6. Divide both sides of the equation by the same term 7. Write $x = $
Solve $\frac{12}{x} = \frac{4}{6}$
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What does it mean to isolate the variable?
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What is the basic technique to isolate a variable?
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What is transposition?
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Algebraic equations can be solved using the Law of Equations.
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How do you isolate a variable in the denominator?
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What are the steps to solve the example $\frac{1}{3}x = 8$?
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What are the steps to solve the example $\frac{3}{x} = 3$?
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What are the rules for solving basic equations with addition?
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What are the rules for solving basic equations with subtraction?
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What are the two rules to remember when multiplying negative numbers?
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Study Notes
Solving for x in Numerators
- Steps to solve for 𝑥 when it is in the numerator include dividing numbers, multiplying all terms by a common value to eliminate fraction denominators, simplifying, and then writing 𝑥 = result.
Example: Solving 14/7 = 𝑥/12
- Initial division gives 2 = 𝑥/12.
- Multiply all terms by 12 to clear the denominator: 24 = 𝑥.
Solving for x in Denominators
- For 𝑥 in the denominator, similarly divide, then clear denominators through multiplication, simplify, and ultimately isolate 𝑥.
Example: Solving 12/𝑥 = 4/6
- Simplification leads to 12 = 2𝑥/3, followed by eliminating fractions and solving for 𝑥 = 18.
Understanding Variable Isolation
- Isolating a variable entails rearranging an algebraic equation to have the variable of interest on one side and all other terms on the opposite side.
Basic Technique for Isolation
- To isolate a variable, perform the same operation on both sides of the equation multiple times as necessary until the variable is alone.
Definition of Transposition
- Transposition is a strategy for isolating a variable, letting one side of the equation contain just that variable.
Law of Equations
- This law asserts that any operation applied to one side of an equation must also be performed on the opposite side to maintain equality.
Isolating Variables in the Denominator
- To isolate variables in the denominator, cross-multiply and then collect like terms.
Example: 1/3𝑥 = 8
- Cross multiplication results in 24𝑥 = 1, leading to 𝑥 = 1/24 after division.
Example: 3/𝑥 = 3
- Cross multiplying yields 3𝑥 = 3, resulting in 𝑥 = 1 after division by 3.
Practice Questions for Isolation
- Several practice equations are provided for students to isolate 𝑥, such as 8/𝑥 + 1 = 4/3 and 4 - 3𝑥 = 40.
Addition Rules in Equation Solving
- For equations involving addition, subtract the constant from both sides to isolate 𝑥. This applies to both forms of addition equations.
Subtraction Rules in Equation Solving
- Equations with subtraction involve adding the variable to both sides first, then subtracting the remaining term from both sides to isolate 𝑥.
Rules for Multiplying Negative Numbers
- Multiplying two negative numbers results in a positive value, while multiplying one negative number yields a negative result.
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Description
This quiz focuses on solving for the variable x when it appears in the numerator and denominator. Students will learn the steps to isolate x, manipulations needed to clear denominators, and practical examples for better understanding. Prepare to apply algebraic techniques to different scenarios involving x!