Math Problem Statement

Solve the system of equations: 2x + 4y - z = 9 -3x + y + z = 14 5x + 2y + z = -6

Solution

The problem involves solving a system of linear equations:

2x+4yz=92x + 4y - z = 9 3x+y+z=14-3x + y + z = 14 5x+2y+z=65x + 2y + z = -6

Steps to Solve the System:

  1. Write the equations in standard form: Equation 1: 2x+4yz=92x + 4y - z = 9
    Equation 2: 3x+y+z=14-3x + y + z = 14
    Equation 3: 5x+2y+z=65x + 2y + z = -6.

  2. Eliminate zz by adding or subtracting equations:

    • From Equation 2 and Equation 3:
      Add Equation 2 (3x+y+z=14-3x + y + z = 14) and Equation 3 (5x+2y+z=65x + 2y + z = -6):
      (3x+y+z)+(5x+2y+z)=14+(6)(-3x + y + z) + (5x + 2y + z) = 14 + (-6)
      Simplify: 2x+3y+2z2z=82x + 3y + 2z - 2z = 8:
      2x+3y=8(New Equation A)2x + 3y = 8 \quad \text{(New Equation A)}

    • From Equation 1 and Equation 3:
      Subtract Equation 3 from Equation 1:
      (2x+4yz)(5x+2y+z)=9(6)(2x + 4y - z) - (5x + 2y + z) = 9 - (-6)
      Simplify:
      3x+2y2z+2z=15-3x + 2y - 2z + 2z = 15:
      3x+2y=15(New Equation B).-3x + 2y = 15 \quad \text{(New Equation B)}.

  3. Solve the reduced system (2x+3y=82x + 3y = 8 and 3x+2y=15-3x + 2y = 15):

    • Multiply Equation A by 3 and Equation B by 2 to eliminate xx:
      3(2x+3y)=3(8)    6x+9y=243(2x + 3y) = 3(8) \implies 6x + 9y = 24
      2(3x+2y)=2(15)    6x+4y=302(-3x + 2y) = 2(15) \implies -6x + 4y = 30

    • Add the two equations:
      (6x+9y)+(6x+4y)=24+30(6x + 9y) + (-6x + 4y) = 24 + 30
      Simplify:
      13y=54    y=5413.13y = 54 \implies y = \frac{54}{13}.

    • Substitute y=5413y = \frac{54}{13} into Equation A:
      2x+3(5413)=82x + 3\left(\frac{54}{13}\right) = 8
      Simplify:
      2x+16213=82x + \frac{162}{13} = 8
      2x=816213=1041316213=5813.2x = 8 - \frac{162}{13} = \frac{104}{13} - \frac{162}{13} = -\frac{58}{13}.
      x=2913.x = -\frac{29}{13}.

  4. Find zz using any original equation:
    Substitute x=2913x = -\frac{29}{13} and y=5413y = \frac{54}{13} into Equation 2 (3x+y+z=14-3x + y + z = 14):
    3(2913)+5413+z=14-3\left(-\frac{29}{13}\right) + \frac{54}{13} + z = 14
    Simplify:
    8713+5413+z=14\frac{87}{13} + \frac{54}{13} + z = 14
    14113+z=14\frac{141}{13} + z = 14
    z=1414113=1821314113=4113.z = 14 - \frac{141}{13} = \frac{182}{13} - \frac{141}{13} = \frac{41}{13}.

Final Solution:

x=2913,  y=5413,  z=4113.x = -\frac{29}{13}, \; y = \frac{54}{13}, \; z = \frac{41}{13}.

Would you like further clarification or step-by-step verification?

5 Follow-Up Questions:

  1. How can systems of equations be solved using matrix methods?
  2. What are the benefits of elimination over substitution in solving systems?
  3. Can graphing always determine solutions for linear systems? Why or why not?
  4. How would the solution change if one equation were inconsistent?
  5. How

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Systems of Equations

Formulas

Substitution Method
Elimination Method

Theorems

Properties of Linear Systems
Solution Uniqueness for Linear Systems

Suitable Grade Level

Grades 9-12