Math Problem Statement

BENAR SALAH z1 + z2 = 2 + i z1 x z2 = 2 + 4i z2 + z3 = 1 + i z1 - z3 = 1 + i z1 + z2 = 2 + i z1 x z2 = 2 + 4i z2 + z3 = 1 + i z1 - z3 = 1 + i

Solution

Based on the information you've provided, we will assume that z1z_1, z2z_2, and z3z_3 are complex numbers, and we are asked to verify the equations.

Let's summarize the given equations for clarity:

  1. z1+z2=2+iz_1 + z_2 = 2 + i
  2. z1×z2=2+4iz_1 \times z_2 = 2 + 4i
  3. z2+z3=1+iz_2 + z_3 = 1 + i
  4. z1z3=1+iz_1 - z_3 = 1 + i

We will attempt to verify the truth of these equations by solving for z1z_1, z2z_2, and z3z_3.

Step 1: Solving for z1z_1 and z2z_2 from equations 1 and 2

From the first equation: z1+z2=2+iz_1 + z_2 = 2 + i

From the second equation: z1×z2=2+4iz_1 \times z_2 = 2 + 4i

This forms a system of equations. To solve for z1z_1 and z2z_2, we can express one variable in terms of the other and substitute it.

Step 2: Solving for z3z_3

From the third equation: z2+z3=1+iz_2 + z_3 = 1 + i This allows us to express z3z_3 in terms of z2z_2: z3=1+iz2z_3 = 1 + i - z_2

We also have the fourth equation: z1z3=1+iz_1 - z_3 = 1 + i Substituting z3z_3 into this equation gives: z1(1+iz2)=1+iz_1 - (1 + i - z_2) = 1 + i

Conclusion:

We can solve these equations using algebraic methods to verify whether the given equalities hold. However, without further clarification or the specific values for z1z_1, z2z_2, and z3z_3, I cannot definitively say whether they are true or false at this stage. Let me know if you'd like to proceed with solving for these variables step by step.


Would you like to continue solving this system, or do you have further details for the values of z1z_1, z2z_2, and z3z_3?

Here are 5 related questions you might want to explore:

  1. How do we solve systems of equations involving complex numbers?
  2. What methods can be used to factor or expand complex number expressions?
  3. How can the modulus and argument of a complex number be found?
  4. What are the properties of complex number multiplication and addition?
  5. How do we visualize complex numbers geometrically?

Tip: When working with complex numbers, remember that addition and multiplication follow similar rules to real numbers, but you must keep track of the imaginary unit ii, where i2=1i^2 = -1.

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Algebraic Systems of Equations

Formulas

z1 + z2 = 2 + i
z1 × z2 = 2 + 4i
z2 + z3 = 1 + i
z1 - z3 = 1 + i

Theorems

Properties of Complex Number Addition and Multiplication

Suitable Grade Level

Grades 10-12