Solving a System of Linear Equations with Substitution: 2x−4y+z=7, −x+y−z=−4, x−2y=3

Math Problem Statement

⎧⎪⎨⎪⎩2x−4y+z=7−x+y−z=−4x−2y=3

We are given a system of three linear equations:

2x - 4y + z = 7 \\ -x + y - z = -4 \\ x - 2y = 3 \end{cases}$$ ### Step 1: Solve the third equation for $$x$$ The third equation is: $$x - 2y = 3$$ Solving for $$x$$: $$x = 2y + 3$$ ### Step 2: Substitute $$x = 2y + 3$$ into the first and second equations #### Substituting into the first equation: $$2(2y + 3) - 4y + z = 7$$ Simplifying: $$4y + 6 - 4y + z = 7 \quad \Rightarrow \quad z = 1$$ #### Substituting into the second equation: $$-(2y + 3) + y - z = -4$$ Substitute $$z = 1$$: $$-(2y + 3) + y - 1 = -4$$ Simplifying: $$-2y - 3 + y - 1 = -4 \quad \Rightarrow \quad -y - 4 = -4 \quad \Rightarrow \quad y = 0$$ ### Step 3: Substitute $$y = 0$$ into $$x = 2y + 3$$ Substitute $$y = 0$$ into $$x = 2y + 3$$: $$x = 2(0) + 3 = 3$$ ### Step 4: Conclusion The solution to the system is: $$x = 3, \quad y = 0, \quad z = 1$$ Would you like further details or have any questions? ### Related Questions: 1. How do you solve a 3x3 system of equations using matrix methods? 2. Can you solve a system of equations graphically? 3. How do substitution and elimination methods compare for solving systems of equations? 4. What are some applications of systems of equations in real life? 5. How can systems of equations be solved using Gaussian elimination? **Tip:** Always check your solution by substituting the values of variables back into the original equations!

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Linear Algebra
Systems of Equations
Substitution Method

x = 2y + 3
Substitution into linear equations

Substitution Theorem
Solving Systems of Linear Equations

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