Math Problem Statement
Find x, y, z, and w if c x + y 2x − 3y z − w z + 2wd = c 4 −7 −6 6 d
Solution
We are given a matrix equation in the form:
x + y & 2x - 3y \\ z - w & z + 2w \end{pmatrix} = \begin{pmatrix} 4 & -7 \\ -6 & 6 \end{pmatrix}$$ To solve for $$x$$, $$y$$, $$z$$, and $$w$$, we can equate corresponding elements of the matrices. This gives us the following system of equations: 1. $$x + y = 4$$ 2. $$2x - 3y = -7$$ 3. $$z - w = -6$$ 4. $$z + 2w = 6$$ Now, let's solve this system of equations. ### Step 1: Solve for $$x$$ and $$y$$ From the first two equations: 1. $$x + y = 4$$ 2. $$2x - 3y = -7$$ We'll solve this system by substitution or elimination. ### Step 2: Solve for $$z$$ and $$w$$ From the last two equations: 3. $$z - w = -6$$ 4. $$z + 2w = 6$$ We'll solve this system as well. I'll start by solving for $$x$$, $$y$$, $$z$$, and $$w$$.The solutions for the variables are: - $$x = 1$$ - $$y = 3$$ - $$z = -2$$ - $$w = 4$$ Would you like any further details or clarifications? Here are 5 related questions for practice: 1. Solve for $$a$$ and $$b$$ if $$a + 2b = 10$$ and $$3a - b = 5$$. 2. If $$4x + y = 12$$ and $$2x - 3y = -7$$, solve for $$x$$ and $$y$$. 3. Find $$z$$ and $$w$$ when $$z + w = 8$$ and $$z - 2w = -1$$. 4. Solve the system: $$5p + q = 7$$ and $$2p - 4q = -10$$. 5. What is the solution for $$x$$ and $$y$$ if $$3x + y = 9$$ and $$x - 2y = 3$$? **Tip:** For solving systems of linear equations, substitution and elimination methods are effective for reducing the system into a simpler form.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Equations
Matrices
Formulas
Matrix Equality
Substitution and Elimination Methods
Theorems
Matrix Equality Theorem
Suitable Grade Level
Grades 10-12