Math Problem Statement

  1. (a)
Solve the following system of equations: (a) x + y + z = 2, 4x + 2y + z = 1, 9x + 3y + z = 8.

Solution

To solve part (a) of the system of equations, let's rewrite the equations for clarity:

  1. x+y+z=2x + y + z = 2
  2. 4x+2y+z=14x + 2y + z = 1
  3. 9x+3y+z=89x + 3y + z = 8

Step 1: Eliminate zz from two pairs of equations.

We subtract the first equation from the second and third equations:

From equation (2) - equation (1): (4x+2y+z)(x+y+z)=12(4x + 2y + z) - (x + y + z) = 1 - 2 3x + y = -1 \tag{4}

From equation (3) - equation (1): (9x+3y+z)(x+y+z)=82(9x + 3y + z) - (x + y + z) = 8 - 2 8x + 2y = 6 \tag{5}

Step 2: Solve the system of two equations in two variables (xx and yy).

Equations (4) and (5) are:

  1. 3x+y=13x + y = -1
  2. 8x+2y=68x + 2y = 6

Multiply equation (4) by 2 to align coefficients of yy: 2(3x+y)=2(1)2(3x + y) = 2(-1) 6x + 2y = -2 \tag{6}

Subtract equation (6) from equation (5): (8x+2y)(6x+2y)=6(2)(8x + 2y) - (6x + 2y) = 6 - (-2) 2x=82x = 8 x=4x = 4

Substitute x=4x = 4 into equation (4): 3(4)+y=13(4) + y = -1 12+y=112 + y = -1 y=13y = -13

Step 3: Solve for zz.

Substitute x=4x = 4 and y=13y = -13 into the first equation: x+y+z=2x + y + z = 2 413+z=24 - 13 + z = 2 9+z=2-9 + z = 2 z=11z = 11

Final Solution:

x=4,y=13,z=11x = 4, \, y = -13, \, z = 11

Would you like a detailed explanation of any step or help with part (b)?

Related Questions:

  1. How would this approach change for systems with more variables?
  2. What are alternative methods for solving systems of equations (e.g., matrix methods)?
  3. How can graphing help visualize solutions to such systems?
  4. What is the significance of consistency in a system of equations?
  5. How does elimination compare to substitution in terms of solving efficiency?

Tip:

Always verify solutions by substituting back into the original equations to ensure accuracy!

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Linear Equations
Elimination Method

Formulas

Equations of the form ax + by + cz = d

Theorems

Elimination Method
Consistency in Systems of Equations

Suitable Grade Level

Grades 9-12