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Questions and Answers
What is the result of simplifying -8 × 7 − (16 - 8)^2?
What is the result of simplifying -8 × 7 − (16 - 8)^2?
-120
Evaluate 7a + b if a = 2 and b = -6.
Evaluate 7a + b if a = 2 and b = -6.
8
What is the range of f(x) = 3x if the domain is {-1, 0, 1}?
What is the range of f(x) = 3x if the domain is {-1, 0, 1}?
{-3, 0, 3}
Evaluate f(x) = -4x + 5 if x = -3.
Evaluate f(x) = -4x + 5 if x = -3.
Simplify 7x^2 + 5x + 4x.
Simplify 7x^2 + 5x + 4x.
Simplify 7(2x + y) + 6(x + 5y).
Simplify 7(2x + y) + 6(x + 5y).
Evaluate |26 - r| + 7 if r = 9.
Evaluate |26 - r| + 7 if r = 9.
Simplify 6x(-4y).
Simplify 6x(-4y).
Simplify (17 + 15) / 8.
Simplify (17 + 15) / 8.
Solve -3/4 y = 8/20.
Solve -3/4 y = 8/20.
Solve 8(x - 5) = 12(4x - 1) + 12.
Solve 8(x - 5) = 12(4x - 1) + 12.
A car dealership has 180 cars on their lot. If they increase their inventory by 25%, how many cars will be on the lot?
A car dealership has 180 cars on their lot. If they increase their inventory by 25%, how many cars will be on the lot?
If f(x) = 7 - 2x, find f(3) + 6.
If f(x) = 7 - 2x, find f(3) + 6.
Chaps is beginning an exercise program that calls for 30 push-ups each day for the first week. Each week thereafter, she has to increase her push-ups by 2. Which week will be the first one in which she will do 50 push-ups a day?
Chaps is beginning an exercise program that calls for 30 push-ups each day for the first week. Each week thereafter, she has to increase her push-ups by 2. Which week will be the first one in which she will do 50 push-ups a day?
A line with a slope of -1 passes through points at (2, 3) and (5, y). Find the value of y.
A line with a slope of -1 passes through points at (2, 3) and (5, y). Find the value of y.
How many solutions exist for the system of equations 2x - 3y = 14 and 4x - 6y = 21?
How many solutions exist for the system of equations 2x - 3y = 14 and 4x - 6y = 21?
When solving the following system, what is x? 5x - 12y = 6 and 7x + y = 3.
When solving the following system, what is x? 5x - 12y = 6 and 7x + y = 3.
If 4x + 5y = 6 and 7x + 5y = 3, what is the value of y?
If 4x + 5y = 6 and 7x + 5y = 3, what is the value of y?
Use substitution to solve the system of equations y = -2x and 5x + 3y = 4. What is the solution?
Use substitution to solve the system of equations y = -2x and 5x + 3y = 4. What is the solution?
Use elimination to solve the system of equations x + 3y = -6 and 2x + 3y = -9. What is the solution?
Use elimination to solve the system of equations x + 3y = -6 and 2x + 3y = -9. What is the solution?
If 4 + 7 + 6 = 4 + 7 + 6 + n, what is the value of n?
If 4 + 7 + 6 = 4 + 7 + 6 + n, what is the value of n?
Solve for x. 4(x + 6) = 3(x + 14).
Solve for x. 4(x + 6) = 3(x + 14).
Find the slope of the line y = -2/3 x + 8.
Find the slope of the line y = -2/3 x + 8.
Solve |-2x + 6| = 7.
Solve |-2x + 6| = 7.
Flashcards
Combining Like Terms
Combining Like Terms
Combining terms with the same variable and exponent. For example, 7x² + 5x + 4x simplifies to 7x² + 9x.
Distributive Property
Distributive Property
A rule that says to multiply the term outside the parentheses with each term inside the parentheses. Example: 7(2x + y) + 6(x + 5y) simplifies to 20x + 37y.
Absolute Value
Absolute Value
The distance a number is from zero, always a positive value. For instance, |-5| is 5. Note that the absolute value of zero is zero.
Simplifying Fractions
Simplifying Fractions
The process of simplifying a fraction to its simplest form by dividing both the numerator and denominator by their greatest common factor.
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Solving Linear Equations
Solving Linear Equations
Solving for the value of an unknown variable in an equation. For example, -3/4y = 8/20 results in y = -8/15.
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Algebraic Manipulations
Algebraic Manipulations
Utilizing various techniques like combining like terms, distributive property, and simplifying fractions to manipulate and simplify expressions. This helps in solving equations and understanding the relationship between variables.
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Function Evaluation
Function Evaluation
Finding the output value of a function, given a specific input value. For example, if f(x) = -4x + 5 at x = -3, then f(-3) = 17.
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Function
Function
A collection of ordered pairs that show the relationship between an input (x) and an output (y) value. A function can be represented by a table, graph, or equation.
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Piecewise Function
Piecewise Function
A function that defines different outputs based on different input ranges. For example, a function might have one rule for x values less than 0 and a different rule for x values greater than or equal to 0.
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Sequence
Sequence
A set of numbers in order, where each number is related to the previous number by a specific pattern or rule. For example, 1, 2, 3, 4 represents the first four natural numbers.
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One-to-One Function
One-to-One Function
A function where every input has only one output. This means that no two different inputs have the same output.
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Percentage Increase
Percentage Increase
Changing the value of an original number by a percentage. For instance, increasing 180 cars by 25% results in 225 cars.
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Slope
Slope
Describing the steepness of a line, calculated as the ratio of the vertical change (rise) to the horizontal change (run).
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Equation of a Line
Equation of a Line
Representing a line with an equation. For example, y = -2/3x + 8 is the equation of a line with a slope of -2/3 and a y-intercept of 8.
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Consistent System of Equations
Consistent System of Equations
A system of linear equations where the lines intersect at one point, representing the solution to both equations.
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Inconsistent System of Equations
Inconsistent System of Equations
A system of linear equations where the lines are parallel and never intersect, signifying no solution.
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Solution of a System of Equations
Solution of a System of Equations
A set of points on a graph that satisfy the conditions of both equations in the system.
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Fraction Simplification
Fraction Simplification
A method for simplifying fractions by dividing both the numerator and denominator by their greatest common factor.
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Equation Solving
Equation Solving
Solving for the value of an unknown variable in an equation by using algebraic operations to isolate the variable on one side of the equation.
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Y-intercept
Y-intercept
The point where a graph crosses the y-axis, where x is equal to 0.
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Substitution Method
Substitution Method
Replacing a variable in one equation with its equivalent expression from a second equation.
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Elimination Method
Elimination Method
Eliminating a variable from a system of equations by adding or subtracting the equations together, aiming to solve for the remaining variable.
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Solving Absolute Value Equations
Solving Absolute Value Equations
A method for solving equations that involve absolute values by considering both positive and negative cases.
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Simplification and Evaluation Concepts
- Simplify expressions using basic algebraic operations, e.g., -8 × 7 − (16 - 8)² results in -120.
- Evaluate expressions at given variable assignments, e.g., 7a + b where a=2 and b=-6 results in 8.
- Understand and find the range of a function given its domain, e.g., f(x) = 3x for domain {-1, 0, 1} leads to range {-3, 0, 3}.
Function Evaluation and Algebraic Manipulations
- Calculate function values by substituting inputs, e.g., f(x) = -4x + 5 at x=-3 results in 17.
- Combine like terms in polynomial expressions, e.g., 7x² + 5x + 4x simplifies to 7x² + 9x.
- Apply distributive property for simplification, e.g., 7(2x + y) + 6(x + 5y) simplifies to 20x + 37y.
Absolute Values and Reductions
- Compute absolute values, as in |26 - r| + 7 with r=9 results in 24.
- Simplify algebraic expressions with multiplication, e.g., 6x(-4y) results in -24xy.
- Simplify fractions and basic arithmetic, e.g., (17 + 15) / 8 simplifies to 4.
Solving Equations
- Solve linear equations for one variable, e.g., -3/4y = 8/20 results in y = -8/15.
- Find solutions to equations by isolating terms, e.g., 8(x - 5) = 12(4x - 1) + 12 leads to x = -1.
- Calculate percent increases in real-world contexts, e.g., a dealership increasing 180 cars by 25% results in 225 cars.
Function Application in Sequences
- Evaluate piecewise functions at given points, e.g., with f(x)=7-2x, finding f(3)+6 results in 7.
- Analyze sequences of push-ups with increasing daily goals, determining that the first week to do 50 push-ups is week 11.
Geometry and Solving Systems of Equations
- Evaluate solutions to linear systems, identifying y for points (2, 3) and (5, y) with slope -1, results in y=0.
- Determine the existence of solutions in equations, understanding when a system yields no solutions, e.g., two equations resulting in a contradiction yields 0 solutions.
- Utilize substitution and elimination methods to solve systems of equations, such as (5x + 3y = 4 and y = -2x) leading to the solution (-4, 8).
Characteristics of Lines
- Determine slopes from equations, e.g., slope of y = -2/3x + 8 is -2/3.
- Solve absolute value equations to find multiple solutions, as | -2x + 6 | = 7 leads to solutions {-1/2, 13/2}.
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