Consider the graph of the system of equations:
-
y = 2 x^{2} + 6 x + 1
-
y = x^{2} + 4 x + 4
State the coordinates of the points of intersection.
For the following systems of equations:
Sketch the graphs of both of the equations on the same set of axes.
State the coordinates of the points of intersection.
y = - x^{3}
y = - x
x^{2} + y^{2} = 25
x + y = 1
x - y = 8
x^{2} + y^{2} = 64
y = x^{2}
y = 2 x^{2} - 1
y = x^{2} - 5
y = 3 - x^{2}
y = x^{2} - 4 x + 2
y = - x^{2} + 8
x^{2} - y = 1
x^{2} + y^{2} = 1
x - y^{2} = 7
x^{2} + y^{2} = 49
x^{2} - y = 0
x + y^{2} = 0
x^{2} - y^{2} = 16
x + y = 8
Consider the hyperbola y = \dfrac{3}{x} and the line y = 5:
At how many points will the two graphs intersect?
State the coordinates of the point(s) of intersection.
Consider the hyperbola x y = 4 and the line y = x + 5.
Sketch the graph of the hyperbola x y = 4 and the graph of the line y = x + 5 on the same set of axes.
How many points of intersection are there for the hyperbola x y = 4 and the line \\y = x + 5.
Determine the exact coordinates of the point(s) of intersection.
For the following pairs of parabolas:
Find the x-coordinate(s) of the point(s) of intersection.
Hence state the coordinates of the point(s) of intersection.
y = \left(x - 2\right)^{2}, y = 52 - x^{2}
y = x^{2} - 9 x - 7, y = x^{2} - x + 25
y = 2 x^{2} + 10 x - 7, y = x^{2} + 8 x + 1
y = - 2 x^{2} - 3 x + 7, y = 7 x^{2} + 9 x + 11
y = \left(x - 8\right)^{2} + 5, y = \left(x - 10\right)^{2} + 9
y = 2 x^{2} + 4 x + 5, y = x^{2} - 2 x - 4
Determine the exact coordinates of the point(s) of intersection for the following system of equations:
-
y = x^{4}
-
y = 16 x^{2} - 60
Determine the exact coordinates of the point(s) of intersection for the following systems of equations:
y = x^{3}
y = 10 x^{2} - 25 x
x^{2} + y^{2} = 41
x - y^{2} = - 21
x^{2} + y^{2} = 4
8 x^{2} - 2 y^{2} = 32
2 x^{2} + 3 y^{2} = 5
x - y = 2
x^2+y^2=5
4x^2+4y^2=24
x^{2} + y^{2} = 70
y = - 5
x^{2} + y^{2} = 10
x - y = 4
x^{2} + y^{2} = 5
- 2 x + 3 y = 7
x^{2} + y = 2
2 y = 5 x^{2} - 24
y = x^{2} - 16
y = 16 - x^{2}
y = x^{2}
y = 4 x^{2} - 12
y = x^{2}
y = x^{2} + 6 x
y = x^{2} + 2
y = 6 - x^{2}
Consider the circles x^{2} + y^{2} = 5 and \left(x - 1\right)^{2} + \left(y - 4\right)^{2} = 2.
State the exact radius of x^{2} + y^{2} = 5.
State the exact radius of \left(x - 1\right)^{2} + \left(y - 4\right)^{2} = 2.
Find the distance between the centres of the two circles.
Hence, how many points of intersection do the two circles have?
The parabolas y = c - a x^{2} and y = x^{2} + 2 x + c intersect at the point \left(1, 0\right).
Find the values of a and c.
The sum of the squares of two numbers is 61. The difference of the squares of the two numbers is 11.
Let x and y be the two numbers.
Hence, determine all the solutions for the two numbers, expressing your answers as ordered pairs.
In business, supply and demand functions are used to express the supply or demand of a commodity as a function of its unit price. Market equilibrium is the term that describes when the supply is equal to the demand. Consider the supply and demand functions for a new book bag below, where y is price per unit (in dollars) and x is the number of units supplied (in thousands):
Supply function: y=2x^2
Demand function: y=-4x+3x^2
Find the market equilibrium quantities and the corresponding prices by solving the given system of simultanous equations for x and y. Write the solutions as ordered pairs.
Hence, describe the number of units that should be produced and the corresponding price per unit, for market equilibrium for this commodity.