Algebra 1: Solving Basic Equations Flashcards

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.