Questions and Answers
What does it mean when two lines intersect at one point?
What does it imply if two lines are parallel?
What does it mean if two lines are identical?
What is the absolute value of any number?
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What does getting a false statement like 0=1 mean when solving a system?
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What are two ways to determine the number of solutions in a system?
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In the system of equations 3s-2t=a and -15s+bt=-7, what is the value of a if the system has infinitely many solutions?
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What happens if the slopes of two lines are both defined (not vertical)?
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What is a linear equation?
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What is a system of linear equations?
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What are linear functions?
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What is an expression in mathematical terms?
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What is an equation?
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What is an inequality?
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What is a function?
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Study Notes
Intersections and Solutions
- Two intersecting lines correspond to one solution.
- Parallel lines indicate no solution exists.
- Identical lines lead to infinitely many solutions.
Absolute Values
- The absolute value of any number is always nonnegative.
System of Equations Insights
- A false statement like 0=1 while solving indicates there are no solutions.
- Two approaches to find the number of solutions in a system: graph the lines or attempt to solve them to see if a contradiction arises.
Systems of Equations Example
- For the given equations 3s-2t=a and -15s+bt=-7, if there are infinitely many solutions, the equations must be identical, meaning coefficients must match when one equation is multiplied by a constant. This leads to b=10 and a=7/5.
Line Geometry
- When two lines have defined (non-vertical) slopes, they are parallel if their slopes are identical, and perpendicular if the product of their slopes equals -1.
Linear Equations and Functions
- A linear equation establishes a relationship between two variables that creates a straight line when graphed.
- A system of linear equations comprises multiple linear equations with the same number of variables, dealt with simultaneously.
- Linear functions are characterized by their straight line graphs represented as y=f(x)=a+bx, where x is independent and y is dependent.
Mathematical Definitions
- An expression lacks an equals sign and represents a value.
- An equation includes an equals sign, defining a specific relationship.
- An inequality describes the value(s) that a variable can assume, illustrating the mathematical relationship without strict equality.
- A function uniquely connects the value of one variable to one or more other variables through a defined process.
Studying That Suits You
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Description
Enhance your understanding of algebra with this set of flashcards focused on the Heart of Algebra section of the SAT. Each card defines key concepts, including solutions to systems of equations and the properties of absolute values. Perfect for quick review and memorization.
