Linear Systems in Three or More Variables
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Linear Systems in Three or More Variables
(teacherweb.com)
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Solve using back-substitution.x – 2y + 3z = 9
y + 3z = 5
z = 2
Sub. z = 2 into 2nd equation.
y + 3(2) = 5y + 6 = 5y = -1
Sub. y = -1 and z = 2 into 1st equation.
x – 2(-1) + 3(2) = 9x + 2 + 6 = 9x + 8 = 9x = 1
Answer (x, y, z ) = (1, -1, 2)
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Objective - To solve systems of linear equations in three variables.
Solve.
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Describe all the ways that three planes could intersect in space.
Intersects at a Point
One Solution
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Describe all the ways that three planes could intersect in space.
Intersects at a Line
Infinitely Many Solutions
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Describe all the ways that three planes could intersect in space.
No Solution
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Describe all the ways that three planes could intersect in space.
No Solution
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Solve.
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Solve.
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Solve.
IdentityInfinitely Many Solutions
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In 1998, Cynthia Cooper of the WNBA Houston Comets basketball team was named Team Sportswoman of the Year. Cooper scored 680 points by hitting 413 of her 1-
pt., 2-pt. and 3-point attempts. She made 40% of her 160 3-pt. field goal attempts. How many 1-, 2- and 3-point
baskets did Ms. Cooper make?
x = number of 1-pt. free throws
y = number of 2-pt. field goals
z = number of 3-pt. field goals
x + y + z = 413
x + 2y + 3z = 680
z/160 = 0.4
-x - y - z = -413 x + 2y + 3z = 680
1 y + 2z = 267
z = 64
y + 2(64) = 267 y = 139x + 139 + 64 = 413 x = 210
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Find a quadratic function f(x) = ax2 + bx + c the graph of which passes through the points (-1, 3), (1, 1), and (2, 6).
Plug in each point for x and y.
a(-1)2 + b(-1) + c = 3
a(1)2 + b(1) + c = 1
a(2)2 + b(2) + c = 6
Simplify a – b + c = 3a + b + c = 14a + 2b + c = 6
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Find a quadratic function f(x) = ax2 + bx + c the graph of which passes through the points (-1, 3), (1, 1), and (2, 6).
a – b + c = 3 a + b + c = 1 4a + 2b + c = 6
a – b + c = 3a + b + c = 1
2a + 2c = 4
-2a - 2b - 2c = -2 4a + 2b + c = 6
2a – c = 4
2a – c = 42a + 2c = 4
-2a + c = -42a + 2c = 4
3c = 0c = 0
a – b + 0 = 3a + b + 0 = 1
a – b = 3 a + b = 1
2a = 4 a = 2
2 + b + 0 = 1 b = -1 f(x) = 2x2 – x