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

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