Solving simultaneous equations using the Koala Bear method, which involves changing the subject of both equations to the same expression then equating, solving and back substituting, problem 2 a circle and a parabola....

From mathswithbob

Solving simultaneous equations using the substitution method, problem 2, a circle and hyperbola, Graphical solution at the end....

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Solving simultaneous equations using the Koala Method, make x or y the subject in both equations, equate and solve, then back substitute for the other variable....

From mathswithbob

Solving simultaneous equations algebraically using the substitution method, problem 1, also solved by elimination in another video....

From mathswithbob

Solving simultaneous equations using the Elimination Method, algebraically, finishing with representation of the solution graphically....

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Using substitution to reduce equations to quadratic equations, in this problem 2, the quartic is reduced to a quadratic giving four solutions on solving further quadratics generated on back substitution...

From mathswithbob

Quadratic inequations are a way of looking at quadratic functions ( parabolas, happy or sad ) and trying to work out x values which generate y values which satisfy the inequality. It is the position of the parabola in the number plane, especially if it cu...

From mathswithbob

Continuing with functions and looking at the shapes of the families, exponential, trigonometric, absolute value and logarithmic....

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Solving single absolute value equations is straight forward, replace the absolute value symbols with +/- brackets, then solve the two cases separately and check your solutions by substituting them back into the original equation . Be aware there may be on...

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Functions are Magic, the number x is transformed as if by magic into another number y, formally f(x), The magic is performed by following a pattern. Together when the points (x,y) or
(x,f(x)) are plotted on the number plane and joined the form the graph ...

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How to find the perpendicular distance, D from a point P, to a plane using the vector dot product. You need to planes equation to extract the normal and a point Q on the plane. Construct the free vector PQ, normalise the normal for the unit vector then us...

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The cross product axb is a vector with the magnitude of the parallelogram formed by the vectors a and b and whose direction is given by the right hand rule, basically perpendicular to the plane containing the vectors a and b. Some examples show how we cou...

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The vector dot product is a scalar or number, found algebraically by multiplying the corresponding components and adding them together, it can be positive or negative. Geometrically it is the magnitude or length of the two vectors multiplied together then...

From mathswithbob

True Bearings are measured in degrees in a clockwise direction from the North line. These Bearings rely on all the north lines being parallel creating equal alternate angles, the exercise uses the cosine rule for non right angle triangles to find the dist...

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Finding the power of a matrix A, first finding the eigenvalues from the characteristic polynomial then using Gaussian reduction to find the associated eigenvectors then forming the diagonal matrix D, of eigenvalues and the matrix of eigenvectors C. Now f...

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Finding the eigenvalues and associated eigenvectors for a matrix A.
The matrix is like a deformation which preserves some vectors direction. The process is to form the characteristic polynomial for the matrix, solve for the eigenvalues, use each eigenvalu...

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Using Gaussian reduction to try to find solutions to 3 by 3 (3D planes ) equations, This method can cope with singular situations, with examples of the inconsistent case 0=-2 ( no solution ) and the redundant case 0=0 ( infinite solutions ) with the help ...

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As an alternative to Cramer's rule the matrix method uses the inverse matrix to solve AX=B . To find the inverse first form the cofactor matrix then transpose the cofactor matrix to form the adjoint matrix, finally divide the adjoint matrix by the determi...

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Cramer's Rule solves systems of linear equations with determinants, you need square systems ie same number of equations as unknowns and you also need a non singular coefficient matrix ( det A not zero ).
You need to form the determinants for each unknow...

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How to find the determinants of 3x3 and 4x4 matrices.
Using the cofactor expansions with alternating signs, Looking at some properties of determinants, using row and column reductions to simplify manual calculations of larger determinants, checking our a...

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How do we multiply matrices. Check compatibility, columns of the first matrix must match the rows of the second matrix. If compatible the result is a matrix which has the rows of the first matrix and the columns of the second matrix, note the order of mul...

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Continuing with the mechanics videos, looking at optimal speeds for banked curves, deriving the equations by resolving the forces horizontally & vertically and tangentially & normally. Looking at how friction affects the velocity equation, and trains with...

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The conical pendulum is a continuation of the circular motion in the horizontal plane series of videos, looking at the theory with 3 examples, the normal conical pendulum, the supported weight through a frictionless ring and the rotation outside on the su...

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Non Uniform circular motion, deriving the horizontal and vertical components of acceleration then resolving these components tangentially and radially/normally with a quick example of the 8km/sec velocity to fling us and everything else tangentially into ...

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Mechanics: Uniform Circular Motion, angular and linear velocity, deriving the tangential and normal components of acceleration with some examples....

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Resisted motion with gravity where the resistance is proportional to the velocity squared R=0.02vv. Finding the maximum height and the time to reach the maximum height ( u=60, v=0, on the rising, and the velocity on reaching the ground after falling back ...

From mathswithbob

Resisted motion with gravity, the resistance proportional to the velocity, R=0.02v. The maximum height and time to reach maximum height on the rising, and the velocity on reaching the ground and the time taken to fall back the ground on the falling....

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Resisted motion falling under gravity where the resistance is proportional to the square of the velocity. Finding the velocity as a function of time, the displacement as a function of velocity, the velocity at a particular height ( x=H ) and the terminal ...

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Resisted motion under gravity where the resistance is proportional to the square of the velocity. Finding an expression for the velocity at time t , an expression for the displacement x , and maximum height H....

From mathswithbob

Set up the force equations for the rising case.The resistance is proportional to the velocity, integrating to find velocity and displacement expressions.
Please note at end the displacement
x=1/k(v-u)... should read x=1/k(u-v).....

From mathswithbob

This is a continuation of the volumes by Slices and Shells 1&2. Using slices and shells for the torus ( doughnut shape), rotating the area inside a circle of radius r and centre (a, b) around the x and y axes. The volume calculations require a good knowle...

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The rectangular hyperbola, perpendicular asymptotes, is rotated 45 degrees to give our old favourite xy=c^2. Equations of tangents and normal in Cartesian and parametric forms, the chords and chord of contact and some properties to finish up....

From mathswithbob