Linear Equations and Substitution Method
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Questions and Answers

Which of the following equations represents a system that can be solved by substitution?

  • 2x + 3y = 6 and y = 4x
  • y = 2x + 1 and 3x + y = 5 (correct)
  • y - x = 1 and y = 3x - 2 (correct)
  • 3y + 2 = 5x and 2x - y = 1

    • The equation y = 3x + 3 can be substituted into any linear equation.

    True (A)

    Using substitution, what is the value of x in the system 2x + y = −4 and y = −3x + 2?

    -2

    In the elimination method, you combine equations to eliminate one of the __________.

    <p>variables</p> Signup and view all the answers

    Match the system of equations with their solution method:

    <p>3x - 2y = -6, 2x + y = -4 = Substitution y = -3x + 2, 2x + y = -4 = Elimination y = x + 2, -x + y = 2 = Substitution 5x - 2y = -6, 3x - y = 7 = Elimination</p> Signup and view all the answers

    Which of the following ordered pairs is a solution to the system of equations: x - y = -1 and 2x - y = -5?

    <p>(-4, -3) (C)</p> Signup and view all the answers

    The ordered pair (1, -3) is a solution to the system of equations: 3x + y = 0 and x + 2y = -5.

    <p>False (B)</p> Signup and view all the answers

    What is the second equation of the system in Example 1?

    <p>2x - y = -5</p> Signup and view all the answers

    In the equation x - y = -1, if x = -2, then y must equal _____ to satisfy the equation.

    <p>-1</p> Signup and view all the answers

    Which equation must be satisfied if the ordered pair (0, 0) is substituted into the system: 3x + y = 0 and x + 2y = -5?

    <p>3(0) + 0 = 0 (B)</p> Signup and view all the answers

    Substituting (-2, -1) into the first equation of Example 1 results in a true statement.

    <p>True (A)</p> Signup and view all the answers

    For the pair (−4, −3), what does the second equation evaluate to?

    <p>-5</p> Signup and view all the answers

    Match the ordered pair with its corresponding status as a solution:

    <p>(-2, -1) = Not a solution (-4, -3) = Solution (1, -3) = Not a solution (0, 0) = Not a solution</p> Signup and view all the answers

    What is the range of the relation {(1, 2), (2, 4), (3, 6), (4, 8), (5, 10)}?

    <p>{2, 4, 6, 8, 10} (D)</p> Signup and view all the answers

    The domain of the relation {(1, 2), (2, 4), (3, 6), (4, 8), (5, 10)} is {2, 4, 6, 8, 10}.

    <p>False (B)</p> Signup and view all the answers

    Define what a function is in mathematical terms.

    <p>A function is a relation where each input value (or domain) is associated with exactly one output value (or range).</p> Signup and view all the answers

    The set of ordered pairs defines a relation, where the first components form the ______ and the second components form the ______.

    <p>domain; range</p> Signup and view all the answers

    Which test can be used to determine if a function is one-to-one?

    <p>Horizontal Line Test (B)</p> Signup and view all the answers

    Match the function-related terms with their definitions:

    <p>Domain = Set of all possible input values Range = Set of all possible output values One-to-one function = A function where different inputs always produce different outputs Vertical Line Test = A method to determine if a relation is a function</p> Signup and view all the answers

    What happens to the value of a stock during the dot-com bubble period?

    <p>It skyrocketed (D)</p> Signup and view all the answers

    What does the quadratic formula solve for?

    <p>The roots of quadratic equations.</p> Signup and view all the answers

    Which ordered pair is a solution to the given system of inequalities?

    <p>(-2, 4) (D)</p> Signup and view all the answers

    The ordered pair (3, 1) satisfies both inequalities in the system.

    <p>False (B)</p> Signup and view all the answers

    What is the first step in solving a system of linear inequalities by graphing?

    <p>Graph the first inequality.</p> Signup and view all the answers

    The solution to a system of two linear inequalities is the region where both _______ are true.

    <p>inequalities</p> Signup and view all the answers

    Match the following steps with their descriptions in solving linear inequalities by graphing:

    <p>Step 1 = Shade the side of the boundary line where the inequality is true Step 2 = Graph the second inequality Step 3 = Locate the overlapping shaded region Step 4 = Choose a test point to check the solution</p> Signup and view all the answers

    When testing if an ordered pair is a solution, what must be true?

    <p>It must satisfy both inequalities. (A)</p> Signup and view all the answers

    The region on one side of the boundary line contains all points that make the inequality true.

    <p>True (A)</p> Signup and view all the answers

    What do you do after graphing the first inequality?

    <p>Graph the second inequality.</p> Signup and view all the answers

    What is the domain of any quadratic function?

    <p>All real numbers (B)</p> Signup and view all the answers

    If the leading coefficient 'a' of a quadratic function is positive, the parabola opens downward.

    <p>False (B)</p> Signup and view all the answers

    What determines whether a parabola has a maximum or minimum value?

    <p>The sign of the leading coefficient 'a'.</p> Signup and view all the answers

    The range of the function f(x) = 2x^2 - 6x + 7 is _____ when the parabola opens upwards.

    <p>[k, ∞)</p> Signup and view all the answers

    What is the range of the function f(x) = -5x^2 + 9x - 1?

    <p>(−∞, 2.25] (C)</p> Signup and view all the answers

    Match the following properties of quadratic functions with their descriptions:

    <p>Domain = Set of all real numbers Maximum value = Occurs when 'a' is negative Minimum value = Occurs when 'a' is positive Range for maximum = f(x) ≤ k</p> Signup and view all the answers

    The vertex of a parabola never represents a maximum or minimum value.

    <p>False (B)</p> Signup and view all the answers

    What is the vertex form of the quadratic function f(x) = 2x^2 - 6x + 7?

    <p>(x - 1.5)^2 + 5.5</p> Signup and view all the answers

    What is the domain of the function $f(x) = 9 - x^2$?

    <p>[−3, 3] (D)</p> Signup and view all the answers

    The domain of the function $f(x) = 4 - 5x$ includes all real numbers.

    <p>True (A)</p> Signup and view all the answers

    Evaluate $g(-3)$ if $g(x) = 3x^2 + 2x - 5$.

    <p>16</p> Signup and view all the answers

    The difference quotient is defined as $\frac{f(x + h) - f(x)}{h}$, where $f(x)$ is a _____ function.

    <p>given</p> Signup and view all the answers

    Match the following functions with their evaluations:

    <p>f(-2) = 14 g(3) = 28 g(-3) + f(-1) = 25 f(-3) = 19</p> Signup and view all the answers

    Which inequality describes the solution for $7 - x ≥ 0$?

    <p>x ≤ 7 (D)</p> Signup and view all the answers

    The set of all real numbers less than or equal to 7 is represented as [−∞, 7).

    <p>False (B)</p> Signup and view all the answers

    What is the difference quotient for the function $f(x) = 7 - 4x$?

    <p>f(x+h) - f(x)/h</p> Signup and view all the answers

    Flashcards

    Solution to a system of equations

    A pair of numbers (x, y) that makes both equations in a system of equations true.

    System of equations

    A set of two or more equations that are considered together.

    Verifying a solution

    A process of substituting the values of x and y from an ordered pair into each equation of a system and checking if both equations are true.

    Ordered pair

    An ordered pair (x, y) that represents a specific point on a coordinate plane.

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    Consistent system

    A system of equations that has at least one solution.

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    Inconsistent system

    A system of equations that has no solutions.

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    Dependent system

    A system of equations that has infinitely many solutions.

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    Graphical interpretation of a solution

    The intersection point of the graphs of two equations in a system represents the solution to that system.

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    Substitution Method

    A method of solving a system of equations by substituting the value of one variable from one equation into another equation to eliminate one variable.

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    Elimination Method

    A method of solving a system of equations by adding or subtracting equations to eliminate one variable.

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    Elimination Method with Opposites

    A system of equations where the variables are multiplied by constants and added together. The coefficients of one variable are opposites, so when added, they cancel out.

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    Elimination Method with Same Coefficients

    A system of equations where the coefficients of one variable are the same, so when subtracted, they cancel out.

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    Solution to a system of linear inequalities

    A pair of values (x, y) that satisfies all the inequalities in a system of linear inequalities.

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    Boundary line

    A line that separates the coordinate plane into two regions, one where the inequality is true and the other where it is false.

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    Checking a solution

    The process of determining whether a given ordered pair satisfies all the inequalities in a system.

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    Solution to a linear inequality

    The region on one side of the boundary line that includes all points that make the inequality true.

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    Solution to a system of linear inequalities (graphically)

    The region where the shaded areas of all the inequalities in a system overlap. It contains all the points that satisfy all the inequalities.

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    Solving a system of linear inequalities by graphing

    The process of finding the solution to a system of linear inequalities by graphing each inequality and identifying the overlapping shaded region.

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    Test point

    A point that is used to test whether the shaded region of an inequality is correct.

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    System of linear inequalities

    A system of linear inequalities with two or more inequalities.

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    What is a function?

    A set of ordered pairs where each input value (from the domain) corresponds to exactly one output value (from the range).

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    What is the domain of a function?

    The set of all possible input values for a function.

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    What is the range of a function?

    The set of all possible output values for a function.

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    What is a one-to-one function?

    A function where each output value corresponds to exactly one input value. In other words, no two different inputs produce the same output.

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    What is the vertical line test?

    If a vertical line intersects the graph of a relation at more than one point, then the relation is not a function.

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    What is the horizontal line test?

    If a horizontal line intersects the graph of a function at more than one point, then the function is not one-to-one.

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    What is an inverse function?

    A function that reverses the action of another function. If f(a) = b, then the inverse function, denoted by f⁻¹(b) = a.

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    What is the composition of functions?

    Applying one function to the output of another function.

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    Domain of a function

    The set of all possible input values for a function. It's the range of values for which the function is defined.

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    Radicand

    The expression under the radical symbol in a radical function, often used to determine the domain.

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    Interval Notation: (-"∞", a]

    A set of numbers that are all less than or equal to a certain value. Represented as (-"∞", a], where 'a' is the upper bound.

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    Interval Notation: [a, b]

    A set of numbers that are between two specific values, inclusive of both. Represented as [a, b], where 'a' and 'b' are the lower and upper bounds.

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    Evaluating a function at a point

    The process of substituting a specific value into a function to find its corresponding output.

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    Difference quotient

    A mathematical expression used to represent the slope of a secant line between two points on a function. It's used in Calculus and has significant application in finding the derivative.

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    Polynomial function

    A function that is described by an equation involving one or more constants. These constants determine the function's behavior and shape.

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    Rational function

    A function where the input variable appears in the denominator of the expression.

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    Quadratic Function

    A mathematical expression that represents a curved path, specifically a parabola, which is the graph of a quadratic function.

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    Standard Form of a Quadratic Function

    The standard form of a quadratic function is f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0.

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    Domain of a Quadratic Function

    The domain of a quadratic function is the set of all real numbers, meaning any value can be plugged in for x.

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    Range of a Quadratic Function

    The range of a quadratic function is restricted by the vertex, which is either a minimum or a maximum point on the parabola. The range consists of all y-values greater than or equal to the minimum (if the parabola opens upwards) or less than or equal to the maximum (if the parabola opens downwards).

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    Vertex of a Parabola

    The vertex of a parabola is the point where the function reaches its minimum or maximum value. It is also the point where the parabola changes direction.

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    Finding the x-coordinate of the Vertex

    The x-coordinate of the vertex of a parabola can be found using the formula x = -b / 2a, where a and b are the coefficients of the quadratic function in standard form.

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    Determining if a Parabola Opens Upwards or Downwards

    If the coefficient 'a' in the standard form of a quadratic function is positive, the parabola opens upwards, and the vertex is a minimum point. If 'a' is negative, the parabola opens downwards, and the vertex is a maximum point.

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    Finding the y-coordinate of the Vertex

    The 'y'-coordinate of the vertex can be found by substituting the x-coordinate of the vertex (found using x = -b / 2a) into the quadratic function.

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    Study Notes

    Applied Mathematics Learning Outcomes

    • A student who satisfactorily completes the course should be able to:
      • Solve two-variable linear equations and inequalities, and sketch their graphs.
      • Interpret a series of three simultaneous inequalities of two variables, display them graphically, and determine the solution set.
      • Demonstrate an understanding of the definition of a function and its graph.
      • Solve quadratic, exponential, and logarithmic equations and inequalities.
      • Solve simple real-life problems involving linear, quadratic, and exponential functions (graphically and algebraically).
      • Determine the zeros and the maximum or minimum of a quadratic function, and solve related problems, including those arising from real-world applications.
      • Sketch the graphs of quadratic, exponential, and logarithmic functions.
      • Compare simple and compound interest and relate compound interest to exponential growth.
      • Understand the inverse relationship between exponents and logarithms and use this relationship to solve related problems.
      • Understand basic probability concepts and compute the probability of simple events using tree diagrams, and formulas for permutations and combinations.
      • Understand basic concepts of descriptive statistics, mean, median, mode, and summarize data into tables and graphs (bar charts, histograms, and pie charts).

    General Foundation Program

    • The program includes courses in mathematics, including equations, inequalities, functions, graphs, exponential and logarithmic functions, and statistics.
    • The content is sourced from various websites, including openstax.org.

    Chapter 1: Systems of Linear Equations and Inequalities

    • Learning Outcomes:
      • Determine if an ordered pair is a solution to a system of equations.
      • Solve systems of linear equations by graphing.
      • Solve a system of equations by substitution.
      • Solve a system of equations by elimination.
      • Solve a system of linear inequalities.

    Chapter 2: Functions and their Graphs

    • Learning Outcomes:
      • Determine whether a relation represents a function.
      • Find the domain and range of a function.
      • Find the value of a function.
      • Determine whether a function is one-to-one.
      • Find the inverse function and composition of functions.
      • Solve quadratic equations using the quadratic formula.
      • Sketch the graphs of linear, quadratic functions.

    Chapter 3: Exponential and Logarithmic Functions

    • Learning Objectives:
      • Understand the concepts and properties of exponential and logarithmic functions.
      • Solve problems based on exponential and logarithmic equations.

    Chapter 4: Statistics

    • Learning Outcomes:
      • Understand basic concepts of descriptive statistics
      • Compute basic measures of central tendency.
      • Summarize given data in to tables and graphs.

    Chapter 5: Probability

    • Learning Objectives:
      • Understand basic concepts of mathematical probability.
      • Compute probability of simple events.
      • Compute probability using tree diagrams, permutations, and combinations.

    Chapter 6: Mathematics of Finance

    • Learning Outcomes:
      • Solve financial problems that involve simple interest and compound interest.
      • Find the future value of an annuity and the amount of payments to a sinking fund.
      • Solve problems involving perpetuity, depletion, and capital cost.
      • Solve problems involving stocks and bonds.
      • Compare simple and compound interest, and relate compound interest to exponential growth.

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    Applied Math Book, Sem 1, 2024-25 PDF

    Description

    Test your understanding of linear equations and the substitution method with this quiz. Answer questions related to solving systems of equations and identify solutions based on given pairs. Perfect for high school math students.

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