Systems of Two Linear Equations in Two Variables Video 5
Solving a system of two linear equations using a TI-84 calculator
Problem: Use a calculator to solve the following system of linear equations: \begin{align*} 5y&=3x+15\\ 0.75&x+3.75=1.25y\end{align*}
Solving a system of two linear equations using a TI-84 calculator
Explaining the format and possible solutions for a system of two linear equations
Solving a system of two linear equations using a TI-84 calculator
Explaining the format and possible solutions for a system of two linear equations
Solving a system of linear equations with two variables
Solving a system of two linear equations with two variables using the elimination method
Solving a linear equation algebraically and showing how this relates to the graph
Solving a linear equation algebraically
Solving a linear equation algebraically and showing how this relates to the graph
Discussing how to write the equation of a line through two given points
Graphing a line without using technology
Graphing a line without using technology
Graphing a line without using technology
Writing a function for the value of an item depreciating linearly over time
Finding the solution to a system of linear equations by graphing
Using a TI-84 to solve a system of linear equations in three variables
Using a TI-84 to solve a system of linear equations in three variables
Graphing the solution set of an inequality with two variables
Graphing the solution set of an inequality with two variables
Graphing the solution set of system of linear inequalities
Explaining the definition of logarithmic scale and double-log plots
Using logarithms to transform a power function into a linear function
Review of the limit definition of a derivative and calculating the derivative
Review of the limit definition of a derivative and calculating the derivative
Review of derivatives and tangent lines to functions and vector equations
Review of the limit definition of a derivative and calculating the derivative
Review of derivatives and tangent lines to functions and vector equations
Review of the limit definition of a derivative and calculating the derivative
Derivatives and tangents to curves
Find the Cartesian form and sketch the graph for the parametric equation of a line
Examples finding a volume using disks, washers, and by slicing
Using the limit definition to find derivatives of functions and vector functions
Review of the limit definition of a derivative and calculating the derivative
Review of the limit definition of a derivative and calculating the derivative
Review of derivatives and tangent lines to functions and vector equations
Using the limit definition to find derivatives of functions and vector functions
Using the limit definition to find derivatives of functions and vector functions
Review of the limit definition of a derivative and calculating the derivative
Review of derivatives and tangent lines to functions and vector equations
Review of the limit definition of a derivative and calculating the derivative
Review of derivatives and tangent lines to functions and vector equations
What the derivatives of a function tell us about the shape of its graph
Optimizing values in word problems
Examples finding a volume using disks, washers, and by slicing
Finding the point where two lines in three-dimensional space intersect
Writing a linear equation of profit from a business word problem
Explaining the main three forms for linear equations
Discussing properties of linear functions
Discussing the types of linear graphs including horizontal and vertical lines and their properties
Explaining when a quadratic equation does not have a real solution
Explaining the graph and properties for the parent function of quadratics
Solving equations with even roots and checking the solutions
Explaining the general properties of even and odd root functions
Properties of exponential growth and decay functions
Properties of the exponential function with base e
Explaining how to solve equations with power and exponential terms
Explaining the basic properties of logarithmic functions
Discussing parent functions of rational functions and their properties
Determining the change in y as x increases for a line
Determining the change in x as y decreases for a line
Determining the slope of a line given points containing a variable
Writing the equation of a line that passes through two points
Finding the slope of a line through a pair of points
Writing the equation of a line with an undefined slope and a given point
Writing the equation of a line with a zero slope given a point
Finding the x and y-intercepts for a line
Solving for a constant so a line has a particular x-intercept
Finding linear functions for the cost, revenue, and profit of a business
Finding the linear cost function for a company
Finding the linear revenue function for a company
Finding the linear profit function for a company
Finding a linear demand equation
Finding a linear supply function
Writing a system of equations for a given word problem
Answering questions about a demand and supply word problem
Finding the change in y as x increases for a given line
Writing a linear function representing the depreciation of a car
Writing the augmented matrix for a system of linear equations
Determining if given values are solutions to a system of linear equations
Solving a system of linear equations without using technology
Maximizing profit subject to a set of constraints
Maximizing wages subject to a set of constraints
Using a graph to find specific values of a function
Finding the maximum revenue, maximum profit, and break even quantity for given revenue and cost functions
Calculating the future value of an investment for different compounding periods
Calculating the effective yields of two compound interest rates to determine which is the better investment
Calculating the future value of an investment with a compounded interest rate
An introduction to using the TVM Solver on the TI-84 calculator
Calculating the future value of an annuity and the amount of interest earned
Calculating the quarterly payment to obtain the desired future value of an annuity
Calculating how many years at a given interest rate to save so the future value is the desired amount
Calculating how long it takes to pay off a debt with compound interest and the total interest paid
Calculating how much money a company should set aside to make retirement payments for 25 years
Calculating payments and interest for the mortgage on a house
Calculating the quarterly payments, total payments, and interest earned for a sinking fund
Calculating the payments and total interest of a loan for different lengths of time
Calculating the values in an amortization table for a loan
Calculating the values in an amortization table for a home mortgage and the equity in the home at certain times
Finding the piecewise-defined function for a given graph along with its domain and range
Graphing a piecewise function
Identifying the parent function and transformations for a given graph
Identifying the parent function and transformations for a given graph
Identifying the parent function and transformations for a given graph
Review of limits, continuity, and differentiability
Using the first and second derivative to sketch the graph of the function
Determining concavity and inflection points and sketching the graph
Using Information From the Derivative to Sketch a Graph
Using the Derivative to Sketch the Graph of a Function
Explaining the logarithmic scale and a semilog plot
Using logarithms to transform an exponential function to a linear function
Writing a non-linear function to represent the line graphed on a semilog plot
Explaining the Sandwich (or Squeeze) Theorem with a graphical example
Solving a limit at infinity using the Sandwich (or Squeeze) Theorem
Finding the equation of a line through two points
Finding the equation of a line through two points
Finding the equations of parallel and perpendicular lines
Finding the equation of a parallel line
Identifying the parent function and transformations of a function
Identifying the parent function and transformations of a function
Determining the properties of a rational function and graphing it
Determining the properties of a rational function and graphing it
Graphically verifying if two functions are inverses
Restricting the domain of a function so it is one-to-one and has an inverse
Determining algebraically the inverse of a function
Determining algebraically the inverse of a function
Determining algebraically the inverse of a function
Determining properties of an exponential function
Determining properties of an exponential function
Determining properties of an exponential function
Identifying the graph of an exponential function
Determining properties of an exponential function
Determining the graph of a logarithmic function
Determining the graph of a logarithmic function
Determining the properties of a sine function and graphing it
Determining the properties of a cosine function and graphing it
Writing the equation for a sine function to match a given graph
Writing the equation for a cosine function to match a given graph
Adding and subtracting vectors graphically
Solving a linear system of equations using substitution
Solving a linear system of equations using elimination
Solving a linear system of equations
Finding the properties of a transformed tangent function and graphing it
Using a graph to evaluate a function
Determining if a function is even or odd
Cartesian equations and parametric equations of curves
Derivatives and vectors with some physics applications
Derivatives and vectors with some physics applications
Review of vector functions and parametric equations
Review of limits, continuity, and the Intermediate Value Theorem
Mean Value Theorem and using derivatives to find the shape of curves
Properties and derivatives of inverse trigonometric functions
Evaluating limits of functions
Evaluating Limits of Functions
Continuity of Functions and the Intermediate Value Theorem
Approximation and Newton's Method, and limits and derivatives of exponential functions
Differentials, linear approximations and quadratic approximations
Approximation and Newton's Method, and limits and derivatives of exponential functions
Derivatives and vectors with some physics applications
Properties and derivatives of inverse trigonometric functions
Derivatives and vectors with some physics applications
Using derivatives to find properties of graphs
Using derivatives to find properties of graphs
Finding the distance between a point and a line using scalar projection
Converting parametric equations into a Cartesian equation and graphing
Finding a vector equation for the line through two points
Finding a parallel vector through a point
Finding a Cartesian equation for a parametric equation and graphing it
Finding a line perpendicular to a vector and passing through a point
Using the graph of a function to sketch a graph of its derivative
Finding the derivatives of compositions of functions from their graphs
Determining the properties of a function from the graph of its derivative
Finding a vector equation for the line passing through two points
Examples finding the area bounded by curves
Using the disk, washer, and shell method to find a volume of revolution
Finding volume using using disks, washers and the shell method
Converting parametric equations into a Cartesian equation and graphing
Converting parametric equations into a Cartesian equation and graphing
Converting parametric equations into a Cartesian equation and graphing
Converting parametric equations into a Cartesian equation and graphing
Finding the area of a region bounded by two curves
Cartesian equations and parametric equations of curves
Proving facts about the derivatives of vector functions including the product rule
Derivatives and vectors with some physics applications
Derivatives and vectors with some physics applications
Review of vector functions and parametric equations
Review of limits, continuity, and the Intermediate Value Theorem
Mean Value Theorem and using derivatives to find the shape of curves
Properties of inverse trig functions and the derivative of arctangent
Properties and derivatives of inverse trigonometric functions
Evaluating limits of functions
Evaluating limits of functions and using the Squeeze Theorem
Proving limits using the epsilon-delta definition
Proving a piecewise function and a polynomial are continuous
Continuity of Functions and the Intermediate Value Theorem
Approximation and Newton's Method, and limits and derivatives of exponential functions
Proving facts about differentials including a product rule
Differentials, linear approximations, and quadratic approximations
Examples involving the tangent line to an exponential function and finding the derivative of hyperbolic cosine
Approximation and Newton's Method, and limits and derivatives of exponential functions
Derivatives and vectors with some physics applications
Properties of inverse trig functions and the derivative of arctangent
Properties and derivatives of inverse trigonometric functions
Using derivatives to find properties of graphs
Using derivatives to find properties of graphs
Proving L'Hospital's Rule and using it to evaluate limits
Proving and then applying the Fundamental Theorem of Calculus
Examples finding the area bounded by curves
Using the disk, washer, and shell method to find a volume of revolution
Finding volume using using disks, washers and the shell method
Showing how an equation in three dimensional space represents a plane
Finding an equation of the plane containing two given lines in three-dimensional space
Identifying the equation of a hyperboloid and sketching its graph
Identifying the equation of a cone in three dimensions and sketching its graph
Finding and sketching the domain of a function of two variables
Finding and sketching the domain of a function of two variables
Sketching the graph of a function of two variables
Sketching the graph of a function of two variables
Sketching the level curves of a function of two variables
Sketching the level curves of a function of two variables
Describing the level surfaces of a function of three variables
Using the method of Lagrange multipliers to find the extreme values of a function subject to a constraint
Evaluating a double integral over a given region
Evaluating a double integral over a given region
Evaluating a double integral and sketching the region of integration
Changing the order of integration for a double integral
Evaluating a double integral by reversing the order of integration
Setting up double integrals of Type I and II to give the volume of a solid under a surface
Evaluating a double integral over a circular region by changing to polar coordinates
Evaluating a double integral by changing to polar coordinates
Evaluating a double integral by changing to polar coordinates
Converting a double integral to a double integral in polar coordinates
Writing an iterated integral in polar coordinates that gives the volume of the solid that lies below a paraboloid
Writing an iterated integral in polar coordinates that gives the volume of the solid bounded by a cone
Writing an a triple integral over a given solid as an iterated integral
Evaluating a triple integral over a solid bounded by given surfaces
Writing an iterated integral that gives the volume of a solid
Evaluating a triple integral over a solid bounded by elliptic paraboloids
Writing a triple integral as an iterated integral in cylindrical coordinates
Converting an iterated triple integral into cylindrical and spherical coordinates
Using an iterated integral in spherical coordinates to find the volume of a solid
Evaluating a triple integral for a given solid by writing an iterated integral in spherical coordinates
Evaluating a line integral along a curve in space
Using Python to find the equation of the tangent line to a curve and graphing the result
Using Python to plot an implicit curve and find a tangent line using implicit differentiation
Using Python to numerically estimate a limit, graphically estimate a limit, and find the exact limit
Using Python to find the tangent line to a parametric equation and plot the two graphs
Graphing a piecewise function using Python
Solving a multistep word problem in Python and graphing the resulting function
Explaining the concepts of existence and uniqueness for solutions of systems of equations
Solving a system of equations with three equations and three variables
Solving a two systems of equations
Using the Gauss-Jordan algorithm to solve a system of equations
Solving a system of equations by row reducing the augmented matrix
Solving a matrix equation using the inverse of the matrix
Evaluating iterated integrals over the same region but with different orders of integration
Writing and then solving an iterated integral for a given region
Using double integrals to calculate the weight of a plate given its density and size
Finding the domain of a function with a denominator, square root, and logarithm
Finding the domain of a function with a natural logarithm and denominator
Finding the domain of a function with a square root in the denominator
Finding the domain of a function with a root and a logarithm in the denominator
Finding the domain of a logarithmic function
Finding the domain of even and odd roots
Finding the domain of functions with even and odd roots
Finding the domain of several rational functions
Finding the domain of a rational function
Explaining how to find any holes and vertical asymptotes of a rational function
Finding the domain of a rational function using factoring
Explaining domain restrictions for denominators, even roots, and logarithms
Explaining the standard form for a quadratic equation and the possible number of solutions
Solving quadratic equations with the difference of two squares formula
Solving quadratic equations by factoring
Solving a quadratic equation by factoring
Solving a quadratic equation by factoring
Solving quadratic equations with the quadratic formula and discussing the number of possible solutions
Discussing properties of quadratic functions including the vertex, domain, range, and end behavior
Solving equations with odd radicals
Solving equations with multiple radicals
How to rewrite radical terms as power terms
Explaining how to solve radical equations and then solving example problems
Solving equations with exponential terms
Solving equations with base e exponential terms
Properties of exponents applied to power and exponential expressions
Properties of multiplying and simplifying power and exponential expressions
Properties of exponents applied to power and exponential expressions
Simplifying power and exponential expressions using properties of exponents
Simplifying power and exponential expressions using properties of exponents
Simplifying power and exponential expressions using properties of exponents
Simplifying an expression with both power and exponential functions using properties of exponents
Explaining the terminology and rules of logarithms
Expanding logarithmic expressions using logarithm rules
Condensing a logarithmic expression into a single logarithm
Explaining how to solve logarithmic equations
Solving logarithmic equations with a single logarithm term
Solving logarithmic equations with multiple logarithm terms
Solving logarithmic equations with multiple logarithm terms on both sides
Finding horizontal asymptotes for rational functions
Explaining how to do polynomial long division
Finding horizontal asymptotes for rational functions
Determining the end behavior and horizontal asymptotes for rational functions
Explaining basic algebraic operations for fractions
How to simplify rational expressions
Adding and subtracting rational expressions by finding a common denominator
Explaining how to multiply and divide rational expressions
Performing algebraic operations with rational expressions and simplifying the answer
Solving equations that contain rational expressions
Solving an equation that contains rational expressions
Solving an equation that contains rational expressions
Solving a system of equations by using the Gauss-Jordan algorithm to row reduce the augmented matrix
Solving a system of equations by using the Gauss-Jordan algorithm to row reduce the augmented matrix
Solving a system of equations by using the Gauss-Jordan algorithm to row reduce the augmented matrix
Solving a system of equations by using the Gauss-Jordan algorithm to row reduce the augmented matrix
Solving a system of equations by using the Gauss-Jordan algorithm to row reduce the augmented matrix
Solving a system of equations by using the Gauss-Jordan algorithm to row reduce the augmented matrix
Solving a system of equations by using the Gauss-Jordan algorithm to row reduce the augmented matrix
Solving a system of equation by writing it as a matrix equation and using the matrix inverse
Introducing functions of several variables and level curves
Evaluating and then finding and sketching the domain of a function of two variables
Finding and sketching the domain of a function of two variables
Sketching the level curves for a function of two variables
Sketching the level curves for a function of two variables
Sketching the level curves for a function of two variables
Finding all local extrema and saddle points for a function of two variables
Determining if a list of vertices in a graph is a path, a circuits, an Euler path, an Euler circuit, or none of the above
Explaining that a graph is a collection of vertices and edges
Drawing a graph to depict a bus route
Counting the vertices and edges from a graph
Identifying the adjacent vertices in a graph
Finding two different paths between vertices in a graph
Determining if a list of vertices in a graph is a path, circuit, Euler path, Euler circuit, or none of the above
Determining if a list of vertices in a graph is a path, a circuits, an Euler path, an Euler circuit, or none of the above
Determining if a list of vertices in a graph is a path, a circuits, an Euler path, an Euler circuit, or none of the above
Determining if a list of vertices in a graph is a path, a circuits, an Euler path, an Euler circuit, or none of the above
Determining if a list of vertices in a graph is a path, a circuits, an Euler path, an Euler circuit, or none of the above
Explaining Euler's Theorem on when a graph has an Euler path or Euler circuit
Finding the valence of each vertex in a graph
Explaining Hamiltonian Circuits and paths, the Method of Trees to find them, and the number of Hamiltonian circuits for complete graphs
Determining if a graph has a Hamiltonian path, Hamiltonian Circuit, or neither
Determining if a graph has a Hamiltonian path, Hamiltonian Circuit, or neither
Determining if a graph has a Hamiltonian path, Hamiltonian Circuit, or neither
Determining if a graph has a Hamiltonian path, Hamiltonian Circuit, or neither
Using the method of trees to find all Hamiltonian circuits of a graph starting at a given vertex
Determining if a graph is a complete graph
Determining if a graph is a complete graph
Determining if a graph is a complete graph
Determining if a graph is a complete graph
Finding two spanning trees for a graph
Introducing functions of several variables and level curves
Explaining matrix operations including addition and multiplication
Solving a system of linear equations with three equations and three variables
Determining the solutions of a system of equations depending on the value of the constants
Defining linear independence for a set of vectors
Determining if three vectors are linearly independent
Explanation of graphs for data and which are appropriate for categorical or numerical data