Math Problem Statement
8x+3y−10z= −1
9x+10y+10z= 3
9x+3y−10z= 6
Solution
We are given a system of three linear equations:
Step 1: Represent the system as a matrix
The coefficient matrix and the constant terms can be written as:
8 & 3 & -10 \\ 9 & 10 & 10 \\ 9 & 3 & -10 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} -1 \\ 3 \\ 6 \end{bmatrix}$$ ### Step 2: Solve using elimination or matrix methods We will solve the system using Gaussian elimination. Let's proceed step by step: #### Subtract equation 1 from equation 3: $$(9x + 3y - 10z) - (8x + 3y - 10z) = 6 - (-1)$$ $$x = 7$$ #### Substitute $$x = 7$$ into equations 1 and 2: From equation 1: $$8(7) + 3y - 10z = -1$$ $$56 + 3y - 10z = -1 \quad \Rightarrow \quad 3y - 10z = -57 \tag{4}$$ From equation 2: $$9(7) + 10y + 10z = 3$$ $$63 + 10y + 10z = 3 \quad \Rightarrow \quad 10y + 10z = -60 \quad \Rightarrow \quad y + z = -6 \tag{5}$$ ### Step 3: Solve the two-variable system From equations (4) and (5): 1. $$3y - 10z = -57$$ 2. $$y + z = -6$$ From equation (5): $$y = -6 - z$$. Substitute into equation (4): $$3(-6 - z) - 10z = -57$$ $$-18 - 3z - 10z = -57$$ $$-18 - 13z = -57 \quad \Rightarrow \quad -13z = -39 \quad \Rightarrow \quad z = 3$$ Substitute $$z = 3$$ into $$y = -6 - z$$: $$y = -6 - 3 = -9$$ ### Final Solution: $$x = 7, \, y = -9, \, z = 3$$ ### Verification: Substitute $$x = 7, y = -9, z = 3$$ into all three equations: 1. $$8(7) + 3(-9) - 10(3) = 56 - 27 - 30 = -1$$ (True) 2. $$9(7) + 10(-9) + 10(3) = 63 - 90 + 30 = 3$$ (True) 3. $$9(7) + 3(-9) - 10(3) = 63 - 27 - 30 = 6$$ (True) The solution is correct. ### Questions: 1. How do you solve larger systems of equations with more variables? 2. What are some applications of solving systems of linear equations? 3. Can we use matrix inversion to solve this system? If so, how? 4. What other methods can solve systems of linear equations (e.g., substitution, elimination)? 5. How does Gaussian elimination handle systems with no or infinite solutions? **Tip:** Always verify solutions by substituting them back into the original equations to avoid calculation errors.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Matrix Methods
Formulas
Gaussian Elimination
Substitution Method
Theorems
Fundamental Theorem of Linear Algebra
Suitable Grade Level
Grades 10-12
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