System of Equations: Determine Number of Solutions

Math Problem Statement

כמה פתרונות יש למערכת במשוואות הזאת ?

The system of equations provided is:

2y - z = 4 \\ y - z = 8 \\ 2z = -16 \end{cases}$$ To determine how many solutions this system has, let's analyze it step by step. --- ### Step 1: Simplify the third equation The third equation is: $$2z = -16$$ Divide by 2: $$z = -8$$ --- ### Step 2: Substitute $$z = -8$$ into the other two equations 1. From the first equation: $$2y - z = 4$$ Substitute $$z = -8$$: $$2y - (-8) = 4 \implies 2y + 8 = 4 \implies 2y = -4 \implies y = -2$$ 2. From the second equation: $$y - z = 8$$ Substitute $$y = -2$$ and $$z = -8$$: $$-2 - (-8) = 8 \implies -2 + 8 = 8 \implies 6 = 8$$ --- ### Step 3: Analyze consistency The second equation produces a contradiction ($$6 = 8$$), meaning that the system of equations is **inconsistent**. --- ### Conclusion The system has **no solutions** because it is inconsistent. The equations contradict each other. --- Would you like me to expand on why inconsistencies like this arise or explain a similar example? Here are five questions that might deepen your understanding: 1. How can you identify if a system is inconsistent from its equations? 2. What is the difference between inconsistent, dependent, and independent systems of equations? 3. How would this system look geometrically (e.g., as lines or planes)? 4. What is a quick way to check for contradictions in a system of equations? 5. How do changes in coefficients affect the consistency of a system? **Tip:** Always substitute back into all equations to confirm consistency when solving systems of equations!

Drop file here or Click Here to upload

Systems of Linear Equations
Inconsistency in Equations

Substitution method: Solve for one variable and substitute into another equation
Basic algebraic manipulation: ax + b = c

Consistency and Inconsistency of Linear Systems

Grades 8-10

Identifying Linear Systems with One Solution - Algebra Practice

Determining the Number of Solutions for Systems of Linear Equations

Solving a System of Linear Equations with Infinitely Many Solutions

How to Determine the Number of Solutions for a System of Equations

Solving Systems of Equations: Infinite and No Solution Cases