Math Problem Statement
Solve the matrix equation shown in the image.
Solution
From the image you uploaded, the given matrix equation is:
2 & 4x \\ x + 3y & 5y + 10z \end{bmatrix} - \begin{bmatrix} 8 & 3 \\ 4 & 5 \end{bmatrix} = \begin{bmatrix} 8 & 5 \\ 9 & -15 \end{bmatrix}$$ We can solve this matrix equation step by step. ### Step 1: Subtract the matrices First, subtract the matrix $$\begin{bmatrix} 8 & 3 \\ 4 & 5 \end{bmatrix}$$ from the matrix $$\begin{bmatrix} 2 & 4x \\ x + 3y & 5y + 10z \end{bmatrix}$$: $$\begin{bmatrix} 2 - 8 & 4x - 3 \\ (x + 3y) - 4 & (5y + 10z) - 5 \end{bmatrix} = \begin{bmatrix} 8 & 5 \\ 9 & -15 \end{bmatrix}$$ Simplifying: $$\begin{bmatrix} -6 & 4x - 3 \\ x + 3y - 4 & 5y + 10z - 5 \end{bmatrix} = \begin{bmatrix} 8 & 5 \\ 9 & -15 \end{bmatrix}$$ ### Step 2: Set up the system of equations Now, equate the corresponding elements of the matrices: 1. $$-6 = 8$$ (This equation has no solution and suggests there may be a mistake or contradiction.) 2. $$4x - 3 = 5$$ 3. $$x + 3y - 4 = 9$$ 4. $$5y + 10z - 5 = -15$$ ### Step 3: Solve the system of equations #### For the second equation: $$4x - 3 = 5$$ Add 3 to both sides: $$4x = 8$$ Divide by 4: $$x = 2$$ #### For the third equation: Substitute $$x = 2$$ into the equation $$x + 3y - 4 = 9$$: $$2 + 3y - 4 = 9$$ Simplify: $$3y - 2 = 9$$ Add 2 to both sides: $$3y = 11$$ Divide by 3: $$y = \frac{11}{3}$$ #### For the fourth equation: Substitute $$y = \frac{11}{3}$$ into the equation $$5y + 10z - 5 = -15$$: $$5\left(\frac{11}{3}\right) + 10z - 5 = -15$$ Simplify: $$\frac{55}{3} + 10z - 5 = -15$$ Multiply everything by 3 to eliminate the fraction: $$55 + 30z - 15 = -45$$ Simplify: $$30z + 40 = -45$$ Subtract 40 from both sides: $$30z = -85$$ Divide by 30: $$z = \frac{-85}{30} = \frac{-17}{6}$$ ### Final Answer: - $$x = 2$$ - $$y = \frac{11}{3}$$ - $$z = \frac{-17}{6}$$ Do you need further details or have any questions? --- Here are 5 related questions: 1. How do you solve systems of linear equations with fractions? 2. What happens if there's no solution for one of the matrix elements? 3. How can matrices be used in solving linear systems? 4. What is the difference between element-wise subtraction and matrix subtraction? 5. Can inconsistent matrix equations still provide useful insights? **Tip:** Always double-check the compatibility of the matrix dimensions when performing operations like addition or subtraction!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Operations
System of Linear Equations
Matrix Subtraction
Algebraic Manipulation
Formulas
Matrix Subtraction
Solving Linear Equations
Theorems
Matrix Equality
Linear Equation Solving
Suitable Grade Level
Grades 10-12