Questions and Answers
Translate 'Three times a number plus five equals to eight times that number' into an algebraic expression.
Evaluate the expression: 9-5(8-3)*2+60.
19
What property is stated during each step of an equation?
No example provided
Use the distributive property to simplify: 4(x-2).
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Solve for x in the equation: 7x - 14 = 8 + 2x*5y.
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Does the equation 4 = 5 have a solution?
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What are the domain and range for the relation {(4,2), (7,1), (3,9)}?
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Identify the independent and dependent variables in the equation y = 2x + 5.
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Every x should have only one corresponding y to determine if a relation represents a function.
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Use function notation to evaluate F(x) when x is given. What is F(3) if F(x) = 4x - 5?
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If a vertical line touches the graph at more than one point, it represents a function.
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What is the x-intercept and y-intercept of a graph?
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If a function's graph is a straight line, it is considered linear.
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A function has line symmetry if it can be mirrored over an axis and appears the same.
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When is the graph considered positive or negative?
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What is the average of the numbers 7, 9, 8, 2, 4, 10, 2?
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An equation is linear if it is a first degree equation.
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Solve this ratio equation: x/8 = 2/4.
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Translate the problem: Tom has 4 apples and Tina has 6 oranges. What is their total?
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What is 10% of 100?
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What is the solution to the equation |x+5|=7?
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Solve the following multi-step equation: 2x - 7 = 1.
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What is the outcome of the equation with variables on both sides: 2x - 7 = x + 1?
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How do you solve for x in the literal equation y = 2x - 9v + 6?
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Study Notes
Translating Expressions
- Convert verbal expressions to algebraic forms: "Three times a number plus five equals eight times that number" becomes 3x + 5 = 8x.
- For inequalities: "Six times a number plus one is greater than four" translates to 6x + 1 > 4.
Order of Operations
- Follow the order of operations (PEMDAS/BODMAS) to simplify expressions accurately.
- For 9 - 5(8 - 3) * 2 + 60, evaluate step by step to achieve a final result of 19.
Properties of Equations
- Identify properties utilized during the resolution of equations, although no specific examples are provided.
Distributive Property
- Apply the distributive property to expand expressions. For instance, 4(x - 2) simplifies to 4x - 8.
Solving Equations
- Use order of operations to solve equations with variables. Example: start with 7x - 14 = 8 + 2x * 5y, rearranging to find x = (44y)/5.
Solution Types
- Recognize when an equation has no solution (e.g., 4 = 5) versus when any number is a solution (e.g., 2x = 2x).
Relations Representation
- Express a relation in multiple forms, including tables, graphs, and mappings, despite no specific examples being given.
Domain and Range
- The domain consists of the x-values while the range comprises the y-values.
- For the set {(4, 2), (7, 1), (3, 9)}, the domain is {4, 7, 3} and the range is {2, 1, 9}.
Variables in Relations
- Identify independent (x) and dependent (y) variables in equations. In y = 2x + 5, y depends on x.
Function Criteria
- A relation is a function if every x-value corresponds to one unique y-value.
- The vertical line test is used to confirm a relation's status as a function on a graph.
Function Notation
- Use function notation for evaluation. Example: F(x) = 4x - 5, then F(3) = 8.
Graph Analysis
- Perform the vertical line test to determine if a graph represents a function—only one point of intersection indicates a function.
Intercepts
- Define x-intercepts as points where the graph intersects the x-axis, and y-intercepts as points where it intersects the y-axis.
Linear vs. Non-Linear Functions
- A linear function produces a straight line graph; any deviation (e.g., curved graphs) indicates a non-linear function.
Symmetry in Graphs
- Identify line symmetry if a graph mirrors perfectly over an axis; lack of this mirror image indicates no symmetry.
Graph Behavior
- Positive values are above the x-axis; negative values are below. Monitor where the graph increases or decreases (no examples provided).
End Behavior
- As x increases or decreases, note how y behaves—determines if the function approaches positive or negative infinity.
Solving Equations
- One-step equations (e.g., x + 5 = 9) result in a direct value for x.
- Multi-step equations (e.g., 2x - 7 = 1) involve multiple transformations to isolate x.
Equations with Variables
- When solving equations like 2x - 7 = x + 1, simplify step by step to find the value of x.
Absolute Value Equations
- Solve for x by creating two separate equations when dealing with absolute values. E.g., |x + 5| = 7 leads to x + 5 = 7 and x + 5 = -7.
Ratios and Proportions
- Solve ratios, such as x/8 = 2/4, by cross-multiplying to find x.
Literal Equations
- Rearranging formulas involves isolating a specific variable. Example: from y = 2x - 9v + 6, solve for x.
Word Problems
- Translate verbal descriptions into equations for problem-solving, such as total fruits equation 4 + 6 = 10.
Percent Calculations
- To find a percentage of a number, like 10% of 100, multiply by the decimal equivalent (0.1).
Averages
- Calculate averages by summing values and dividing by the total number of values. In this case, the average of 7 + 9 + 8 + 2 + 4 + 10 + 2 is 7.
Linear Function Identification
- Ensure equations are in standard form ax + by = c. First-degree equations signify linear functions.
Graphing Linear Functions
- To graph a linear function, identify x and y intercepts and connect them, ensuring accurate visualization.
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Test your knowledge with these algebra flashcards focused on translating verbal expressions, using order of operations, and properties of algebraic expressions. Perfect for preparing for your final exam in Algebra. Challenge yourself and reinforce your understanding!
