Math Problem Statement

x+2y-3z= -4 2x+y-3z= 4 solve by elimination method

Solution

To solve the system of equations by the elimination method, let’s start with the system:

x+2y3z=4(1)x + 2y - 3z = -4 \quad (1) 2x+y3z=4(2)2x + y - 3z = 4 \quad (2)

Step 1: Eliminate zz by subtracting Equation (1) from Equation (2).

To eliminate zz, notice that both equations contain 3z-3z with the same coefficient. By subtracting Equation (1) from Equation (2), we can remove zz from the equations:

(2x+y3z)(x+2y3z)=4(4)(2x + y - 3z) - (x + 2y - 3z) = 4 - (-4)

Simplify each side:

2x+y3zx2y+3z=82x + y - 3z - x - 2y + 3z = 8

This simplifies to:

xy=8(3)x - y = 8 \quad (3)

Step 2: Solve Equation (3) for xx in terms of yy:

From Equation (3), we have:

x=y+8x = y + 8

Step 3: Substitute x=y+8x = y + 8 into one of the original equations.

Let’s substitute x=y+8x = y + 8 into Equation (1):

(y+8)+2y3z=4(y + 8) + 2y - 3z = -4

Simplify this:

3y+83z=43y + 8 - 3z = -4

Subtract 8 from both sides:

3y3z=123y - 3z = -12

Divide through by 3:

yz=4(4)y - z = -4 \quad (4)

Step 4: Solve Equation (4) for yy in terms of zz:

From Equation (4), we have:

y=z4y = z - 4

Step 5: Substitute y=z4y = z - 4 back into x=y+8x = y + 8:

Substitute y=z4y = z - 4 into x=y+8x = y + 8:

x=(z4)+8x = (z - 4) + 8

Simplify:

x=z+4x = z + 4

Conclusion:

The solution to this system in terms of zz is:

x=z+4,y=z4x = z + 4, \quad y = z - 4

Therefore, the solution set is all points (x,y,z)(x, y, z) that satisfy:

(x,y,z)=(z+4,z4,z)(x, y, z) = (z + 4, z - 4, z)

If you have a specific value for zz, you can substitute it to find the corresponding values of xx and yy.

Would you like more details or have questions on this method? Here are five follow-up questions to consider:

  1. What happens if we use a different elimination strategy?
  2. How can we verify the solution by substitution?
  3. What is the geometric interpretation of this solution set?
  4. Can we express the solution in a parametric form?
  5. How does elimination differ from substitution for this system?

Tip: Always double-check your solution by plugging it back into the original equations to ensure accuracy.

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Math Problem Analysis

Mathematical Concepts

Algebra
Systems of Linear Equations
Elimination Method

Formulas

Elimination method for solving systems of linear equations
Linear combination of equations

Theorems

Properties of equality in linear systems

Suitable Grade Level

Grades 10-12