Podcast
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 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?
What is x + 5 = 8?
x = 3
Isolate the variable (x) by performing the same operations to ____ ______.
Isolate the variable (x) by performing the same operations to ____ ______.
both sides
What is another term for a solution to an equation?
What is another term for a solution to an equation?
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What is the result of (a/b)(b/a)?
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?
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?
Which of the following is a quadratic equation?
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If AB = 0, what values could A and B be?
If AB = 0, what values could A and B be?
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Most quadratic equations will have ___ possible solutions.
Most quadratic equations will have ___ possible solutions.
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What values does x take for the equation x² - 9 = 0?
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?
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?
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?
What is the result of the equation √x = 5?
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True or False? |x| = x if x ≥ 0.
True or False? |x| = x if x ≥ 0.
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Absolute Value is a number's distance from ____ on a number line.
Absolute Value is a number's distance from ____ on a number line.
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What is the solution for |x| = 4?
What is the solution for |x| = 4?
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How do you solve |x - 10| = 6?
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
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
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.
