Math Problem Statement
Find values for the variables so that the matrices are equal. left bracket Start 2 By 2 Matrix 1st Row 1st Column x plus 3 2nd Column y plus 4 2nd Row 1st Column 7 2nd Column negative 5 EndMatrix right bracket equals left bracket Start 2 By 2 Matrix 1st Row 1st Column 9 2nd Column 5 2nd Row 1st Column 7 2nd Column z EndMatrix right bracket
Solution
We are given two 2x2 matrices and need to find values for , , and such that the matrices are equal. The equation is:
x + 3 & y + 4 \\ 7 & -5 \end{pmatrix} = \begin{pmatrix} 9 & 5 \\ 7 & z \end{pmatrix}$$ For two matrices to be equal, their corresponding elements must be the same. This leads to the following system of equations: 1. $$x + 3 = 9$$ 2. $$y + 4 = 5$$ 3. $$7 = 7$$ (This is always true, so it doesn't provide any new information.) 4. $$-5 = z$$ ### Solving for the variables: 1. From $$x + 3 = 9$$, we solve for $$x$$: $$x = 9 - 3 = 6$$ 2. From $$y + 4 = 5$$, we solve for $$y$$: $$y = 5 - 4 = 1$$ 3. From $$-5 = z$$, we solve for $$z$$: $$z = -5$$ Thus, the values for the variables are: - $$x = 6$$ - $$y = 1$$ - $$z = -5$$ Let me know if you need further explanations or more problems! ### Additional Questions: 1. What is the meaning of matrix equality in linear algebra? 2. How can you use matrix operations to solve systems of equations? 3. Can a matrix equation have multiple solutions? If so, under what conditions? 4. How would you solve for a variable in a matrix equation involving more than two matrices? 5. What is the difference between scalar and matrix addition? ### Tip: Remember that when matrices are equal, their corresponding elements must match exactly!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Equality
System of Linear Equations
Formulas
x + 3 = 9
y + 4 = 5
-5 = z
Theorems
Matrix equality theorem
Suitable Grade Level
Grades 9-11