Questions and Answers
How do you solve an equation like x - 5 = 2?
Add 5 to both sides to isolate x, resulting in x = 7.
How do you solve an equation like x + 6 = 10?
Subtract 6 from both sides to isolate x, resulting in x = 4.
How do you solve an equation like 5x = 35?
Divide both sides by 5 to isolate x, resulting in x = 7.
How do you solve an equation like x/4 = 8?
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How do you solve an equation like 3/5x = 2?
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If x = 1/2z and 3y = 5z, which of the following relationships holds between x and y for each nonzero value of z?
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How do you solve x² = 16?
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How do you solve x³ = 8?
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How do you solve √x = 6?
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How do you solve ₃√x = 2?
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How do you solve r/10 + 4 = 5?
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How do you solve −9 + n/4 = −7?
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How do you solve 1 − r = −5?
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How do you solve x/(−4) − 5 = −8?
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What is the solution for x of the equation b - ax = c?
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Given the equations M = 3 - x and y = N + 4, which of the following expressions is equivalent to M - N written in terms of x and y?
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If x and y are positive integers and x - y = 7, what is the least possible value of xy?
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For positive real numbers x, y, and z such that 5x = 2y and 1/6y = 2/3z, which of the following inequalities is true?
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How do you solve (k - 10) / 2 = -7?
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Study Notes
Isolating Variables
- To isolate a variable, always perform the opposite operation of the number adjacent to it.
- For an equation like x - 5 = 2, add 5 to both sides: x = 7.
- For x + 6 = 10, subtract 6 from both sides: x = 4.
- In 5x = 35, divide both sides by 5 to get x = 7.
Dealing with Fractions
- To solve equations with fractions, multiply by the reciprocal.
- For (3/5)x = 2, multiply both sides by (5/3) to find x = 10/3.
Working with Exponents
- Take the square root to solve x² = 16, yielding x = ±4.
- For x³ = 8, use cube roots to find x = 2.
- Remember that the square root and square cancel each other out: (√x)² = 6² results in x = 36.
- Raise both sides to eliminate the root in ₃√x = 2, resulting in x = 8.
Multi-Step Equations
- Apply the PEMDAS method in reverse: handle operations starting with addition or subtraction, then multiplication or division.
- In the equation r/10 + 4 = 5, subtract 4 and multiply by 10, leading to r = 10.
Solving for Relationships
- For equations involving parameters, manipulate them to find relationships (e.g., x = 1/2z and 3y = 5z) to derive expressions like 10x = 3y.
Understanding Embedded Expressions
- Dissolve fractions by multiplying both sides to clear them.
- For (k - 10) / 2 = -7, multiply by 2, resulting in k - 10 = -14, leading to k = -4.
Inequalities and Positivity
- In inequalities with positive integers (x - y = 7), discern their least product (e.g., x = 8, y = 1).
- For constraints involving positive real numbers, analyze the relationships established by given equations to infer inequalities like z < x < y.
Conceptual Equations
- Equations can be rewritten or transformed to find equivalent expressions, such as M - N expressed in terms of x and y.
Practice Problems
- Engage with various equations and solutions such as n/4 = -7, 1 - r = -5, and check computations involving embedded variables and operations.
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Test your skills in solving basic algebraic equations with these flashcards. Each card presents an equation and walks you through the steps needed to isolate the variable. Perfect for mastering fundamental algebra concepts!