Math Problem Statement
calc:
Solve the system of linear equations: \( y - 2z = -2 \), \( 2x + 3y - 5z = -5 \), \( x + z = 1 \)
Solution
To solve this system of linear equations, let’s analyze and solve each equation step-by-step.
Given system:
y - 2z = -2 \\ 2x + 3y - 5z = -5 \\ x + z = 1 \end{cases}$$ ### Step 1: Solve for $$x$$ from the third equation From the third equation: $$x + z = 1$$ we can rearrange to express $$x$$ in terms of $$z$$: $$x = 1 - z$$ ### Step 2: Substitute $$x = 1 - z$$ into the second equation Substituting $$x = 1 - z$$ into the second equation: $$2(1 - z) + 3y - 5z = -5$$ Expanding and simplifying: $$2 - 2z + 3y - 5z = -5$$ $$3y - 7z = -7$$ So, we have a new equation: $$3y - 7z = -7$$ ### Step 3: Solve for $$y$$ using the first equation The first equation is: $$y - 2z = -2$$ Solving for $$y$$: $$y = -2 + 2z$$ ### Step 4: Substitute $$y = -2 + 2z$$ into the modified second equation Now substitute $$y = -2 + 2z$$ into the equation $$3y - 7z = -7$$: $$3(-2 + 2z) - 7z = -7$$ Expanding: $$-6 + 6z - 7z = -7$$ $$-6 - z = -7$$ $$z = 1$$ ### Step 5: Substitute $$z = 1$$ back into $$x = 1 - z$$ and $$y = -2 + 2z$$ With $$z = 1$$: 1. For $$x$$: $$x = 1 - 1 = 0$$ 2. For $$y$$: $$y = -2 + 2 \cdot 1 = 0$$ ### Solution The solution to the system is: $$(x, y, z) = (0, 0, 1)$$ Would you like more details, or do you have any questions about this solution? Here are some related questions to further explore: 1. How would the solution change if one of the equations were slightly altered? 2. What is the geometric interpretation of this solution in three-dimensional space? 3. How can you verify the solution by substituting back into the original equations? 4. What are some common methods for solving systems of equations, and when are they used? 5. How does this method compare with matrix-based approaches like Gaussian elimination? **Tip:** Always check your solution by substituting it back into the original equations to ensure accuracy.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Substitution Method
Formulas
Substitution method formulas for solving systems of equations
Theorems
None directly, but related to the concepts of consistency in linear systems
Suitable Grade Level
Grade 10-12
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