Math Problem Statement

Solve the following system of equations for all three variables. 8x + y + 4z = 9 4x - 3y + z = 7 8x - 4y + 5z = -2

Solution

To solve this system of equations for xx, yy, and zz, let's start by writing down the equations:

  1. 8x+y+4z=98x + y + 4z = 9
  2. 4x−3y+z=74x - 3y + z = 7
  3. 8x−4y+5z=−28x - 4y + 5z = -2

Step 1: Eliminate one variable

Let's eliminate xx by combining equations.

  1. Subtract Equation 1 from Equation 3: (8x−4y+5z)−(8x+y+4z)=−2−9(8x - 4y + 5z) - (8x + y + 4z) = -2 - 9 This simplifies to: −5y+z=−11⇒z=−11+5y-5y + z = -11 \quad \Rightarrow \quad z = -11 + 5y

  2. Combine Equations 1 and 2:

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Math Problem Analysis

Mathematical Concepts

Algebra
Systems of Linear Equations

Formulas

Elimination method
Substitution method

Theorems

Linear combination of equations
Properties of equality

Suitable Grade Level

Grades 9-12

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