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If: y -x = 7

Then: y = 7+x

If: y = 8/x

Then: xy = 8

Therefore it follows that: x(7+x) = 8

So: 7x+x^2 = 8 => 7x^2 -8 = 0

Using the quadratic equation formula: x = -8 or x = 1

By substitution the line intercepts the curve at (-8, -1) and (1, 8)

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They are (-8, -1) and (1, 8).

Q: What are the points of intersection when the line y -x equals 7 spans the curve y equals 8 over x on the Cartesian plane?

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They work out as: (-3, 1) and (2, -14)

If: y = x2-x-12 Then points of contact are at: (0, -12), (4, 0) and (-3, 0)

The two solutions are (x, y) = (-0.5, -sqrt(3.5)) and (-0.5, sqrt(3.5))

Equations: x -y = 2 and x^2 -4y^2 = 5 By combining the equations into a single quadratic equation in terms of y and solving it: y = 1/3 or y = 1 By means of substitution the points of intersection are at: (7/3, 1/3) and (3, 1)

You get a curve. If you join them along the shortest [Euclidean] distance between them, you get a straight line.

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They work out as: (-3, 1) and (2, -14)

Straight line: 3x-y = 5 Curved parabola: 2x^2 +y^2 = 129 Points of intersection works out as: (52/11, 101/11) and (-2, -11)

If: y = x2-x-12 Then points of contact are at: (0, -12), (4, 0) and (-3, 0)

The two solutions are (x, y) = (-0.5, -sqrt(3.5)) and (-0.5, sqrt(3.5))

14

You get a curve. If you join them along the shortest [Euclidean] distance between them, you get a straight line.

Equations: x -y = 2 and x^2 -4y^2 = 5 By combining the equations into a single quadratic equation in terms of y and solving it: y = 1/3 or y = 1 By means of substitution the points of intersection are at: (7/3, 1/3) and (3, 1)

The intersection of the individual graphs. In the simplest case, the graph for each equation consists of a line (or some curve); the intersection is the points where the lines or curves meet.

If: x-2y = 1 and 3xy-y2 = 8 Then: x =1+2y and so 3(1+2y)y-y2 = 8 => 3y+5y2-8 = 0 Solving the quadratic equation: y = 1 or y = -8/5 Points of intersection by substitution: (3, 1) and (-11/5, -8/5)

the equilibrium price of a good or service

If: y = 10x -12 and y = x^2 +20x +12 Then: x^2 +20x +12 = 10x -12 Transposing terms: x^2 +10x +24 = 0 Factorizing: (x+6)(x+4) = 0 => x = -6 or x = -4 Points of intersection by substitution are at: (-6, -72) and (-4, -52)

If: y = -8 -3x and y = -2 -4x -x^2 Then: -8 -3x = -2 -4x - x^2 Transposing terms: x^2 +x -6 = 0 Factorizing: (x-2)(x+3) = 0 => x = 2 or x = -3 Points of intersection by substitution are at: (2, -14) and (-3, 1)