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Advanced Algebra Notes Section 3.4: Solve Systems of Linear Equations in Three Variables

Advanced Algebra Notes Section 3.4: Solve Systems of Linear Equations in Three Variables. l inear equation in 3 variables. A ___________________________ x, y, and z is an equation of the form ax + by + cz = d where a, b, and c are not all zero.

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Advanced Algebra Notes Section 3.4: Solve Systems of Linear Equations in Three Variables

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  1. Advanced Algebra Notes Section 3.4: Solve Systems of Linear Equations in Three Variables linear equation in 3 variables • A ___________________________ x, y, and z is an equation of the form ax + by + cz = d where a, b, and c are not all zero. • The following is an example of a ________________________ in 3 variables: • 2x + 5y – z = -7 • x – 3y + z = 10 • 9x + y – 4z = -1 • The solution to a system of equation in 3 variables is called an ______________ (x, y, z) whose coordinates make all 3 equations true. • The _______ of a linear system in 3 variables is a plane in three-dimensional space. • The intersection of the 3 planes determines the number of solutions. • The planes intersect in a ____________. • The planes intersect in a ______ or are the _____________. (Infinitely Many Solutions) • The planes have __________________ of intersection. ( No Solution) system of linear equations ordered triple graph single point line same plane no common point

  2. Steps: • 1. Rewrite 2 of the equations in 3 variables as equations in 2 variables using substitution • or elimination. • 2. Solve those 2 equations for both variables like you were taught in section 3.2. • 3. Once you get the values from step 2, substitute them in to one of the original • equations and solve for the 3rd value. Then write your ordered triple. • ** If the variables disappear and you get a false statement -3 = 0, then the system has no • solutions. • ** If the variables disappear and you get a true statement 0 = 0, then the system has • infinitely many ordered triple solutions.

  3. Examples: • 1. 2x – y + 6z = -4 • 6x + 4y – 5z = -7 • -4x – 2y + 5z = 9 -y = -2x – 6z – 4 y = 2x + 6z + 4 6x + 4(2x + 6z + 4) – 5z = -7 6x + 8x + 24z + 16 – 5z = -7 14x + 19z = - 23 -4x – 2(2x + 6z + 4) + 5z = 9 -4x – 4x – 12z – 8 + 5z = 9 -8x – 7z = 17

  4. 2. 3x + y – 2z = 10 • 6x – 2y + z = -2 • x + 4y + 3z = 7

  5. 3. x + y – z = 2 • 3x + 3y – 3z = 8 • 2x – y + 4z = 7

  6. 4. x + y + z = 6 • x – y + z = 6 • 4x + y + 4z = 24

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