Questions and Answers
What is the term for the rule that states what you do to one side of the equation, you must do to the other side?
What is x + 5 = 8?
x = 3
Isolate the variable (x) by performing the same operations to ____ ______.
both sides
What is another term for a solution to an equation?
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What is the result of (a/b)(b/a)?
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What equation do you solve for x in (3x/4) + 1/2 = x/3?
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Which of the following is a quadratic equation?
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If AB = 0, what values could A and B be?
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Most quadratic equations will have ___ possible solutions.
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What values does x take for the equation x² - 9 = 0?
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What happens if (b² - 4ac) < 0 in a quadratic equation?
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For the system of equations, what does 3x + 2y = 9 and 6x + 4y = 11 imply?
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What is the result of the equation √x = 5?
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True or False? |x| = x if x ≥ 0.
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Absolute Value is a number's distance from ____ on a number line.
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What is the solution for |x| = 4?
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How do you solve |x - 10| = 6?
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Study Notes
Basic Equation Solving
- Basic concept involves finding the value of variables that satisfy equations.
- The Golden Rule: Any operation applied to one side of an equation must also be applied to the other side to maintain equality.
Eliminating Fractions
- Multiplying both sides of an equation by a common denominator can eliminate fractions.
- Example: For the equation (3x/4) + 1/2 = x/3, finding a common denominator will simplify the solving process.
Quadratic Equations
- A quadratic equation is expressed in the standard form: ax² + bx + c = 0, where a cannot be zero.
- Common examples include equations like 3x² - 2x - 1 = 8 and x² - 9 = 0.
- Solutions to quadratic equations can be found using methods such as factoring, completing the square, or applying the quadratic formula.
Number of Solutions in Quadratics
- Quadratic equations can have two, one, or no real solutions depending on the value of the discriminant (b² - 4ac).
- If b² - 4ac < 0, no real solutions exist; if b² - 4ac ≥ 0, solutions exist.
Solving Systems of Linear Equations
- Systems of equations with two variables can be solved using substitution or elimination methods.
- The substitution method involves solving one equation for one variable and substituting this value into the other equation.
- The elimination method requires aligning coefficients to eliminate one variable and solving for the other.
Unique Solutions in Systems
- A system can have no solutions (inconsistent), one solution (independent), or infinitely many solutions (dependent).
- Identical equations indicate infinitely many solutions, while contradictory equations show no solutions.
Equations with Square Roots
- To solve an equation involving a square root, square both sides to eliminate the root.
- Example: For √(x-2) = 3, squaring both sides gives x - 2 = 9, leading to x = 11.
Extraneous Roots
- Solutions obtained may not satisfy the original equation, known as extraneous roots; always verify solutions against the original equation.
Powers and Bases
- If b^x = b^y, then x must equal y provided b is non-zero.
- This property does not hold true for bases of 0 or 1.
Absolute Value Equations
- The absolute value |x| represents the distance from zero; hence |x| = a means x could be a or -a.
- To solve an absolute value equation, consider both possible cases defined by the absolute value.
Equations and Absolute Values
- A procedure for solving includes setting up two separate equations based on the absolute value definition and solving each case individually.
- Validity checks are necessary for solutions derived from absolute value equations.
Studying That Suits You
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Description
Test your understanding of basic equation solving concepts through flashcards covering videos 15-26. This quiz includes definitions of key terms and examples to reinforce your learning. Ideal for students who want to grasp essential algebra techniques.
