\begin{pmatrix} a(1) + b(3) & a(2) + b(4) \\ c(1) + d(3) & c(2) + d(4) \end{pmatrix} = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}.

["Solving the Linear Equation of a Matrix: A Step-by-Step Guide to [ \begin{pmatrix} a(1) + b(3) & a(2) + b(4) \ c(1) + d(3) & c(2) + d(4) \end{pmatrix} = \begin{pmatrix} 5 & 6 \ 7 & 8 \end{pmatrix} ]", "Linear equations aren’t just abstract math — they’re essential tools in engineering, computer science, and data modeling. One such equation we often encounter is a matrix equation involving unknowns expressed through linear combinations:", "[\n\begin{pmatrix} a(1) + b(3) & a(2) + b(4) \ c(1) + d(3) & c(2) + d(4) \end{pmatrix} = \begin{pmatrix} 5 & 6 \ 7 & 8 \end{pmatrix}\n]", "In this article, we’ll break down how to interpret, solve, and analyze this matrix equation, helping you understand both the algebraic structure and practical implications.", "---", "### What Does the Matrix Equation Represent?", "The equation sets a ( 2 \ imes 2 ) matrix equal to a known constant matrix:", "[\n\begin{pmatrix} a + 3b & 2a + 4b \ c + 3d & 2c + 4d \end{pmatrix} = \begin{pmatrix} 5 & 6 \ 7 & 8 \end{pmatrix}\n]", "This means that each entry of the linear combination matrices must match the corresponding entry in the constant matrix:", "[\n\begin{align}\na + 3b &= 5 \quad \ ext{(top-left)} \\n2a + 4b &= 6 \quad \ ext{(top-right)} \\nc + 3d &= 7 \quad \ ext{(bottom-left)} \\n2c + 4d &= 8 \quad \ ext{(bottom-right)}\n\end{align}\n]", "---", "### Analyzing the System of Equations", "Each row forms a system of linear equations in two variables. We solve them row-by-row.", "#### Top Row: ( a + 3b = 5 ) and ( 2a + 4b = 6 )", "We already can express ( a ) in terms of ( b ):", "[\na = 5 - 3b\n]", "Substitute into the second equation:", "[\n2(5 - 3b) + 4b = 6 \implies 10 - 6b + 4b = 6 \implies 10 - 2b = 6 \implies -2b = -4 \implies b = 2\n]", "Then substitute back to find ( a ):", "[\na = 5 - 3(2) = 5 - 6 = -1\n]", "#### Bottom Row: ( c + 3d = 7 ) and ( 2c + 4d = 8 )", "Similarly, solve for ( c ):", "[\nc = 7 - 3d\n]", "Substitute into the second equation:", "[\n2(7 - 3d) + 4d = 8 \implies 14 - 6d + 4d = 8 \implies 14 - 2d = 8 \implies -2d = -6 \implies d = 3\n]", "Find ( c ):", "[\nc = 7 - 3(3) = 7 - 9 = -2\n]", "---", "### The Solution", "Putting together the values:", "[\n\begin{align}\na &= -1 \\nb &= 2 \\nc &= -2 \\nd &= 3\n\end{align}\n]", "The matrix equation holds true because substituting back:", "[\n\begin{pmatrix} (-1)(1) + 2(3) & (-1)(2) + 2(4) \ (-2)(1) + 3(3) & (-2)(2) + 3(4) \end{pmatrix} = \begin{pmatrix} 5 & 6 \ 7 & 8 \end{pmatrix}\n]", "Verify:", "- ( -1 + 6 = 5 ), ( -2 + 8 = 6 )\n- ( -2 + 9 = 7 ), ( -4 + 12 = 8 ) ✅", "---", "### Why Does This Matrix Representation Matter?", "In applied contexts, such equations model transformations, coordinate mapping, or linear relationships between variables — common in computer graphics, econometrics, and solving systems of equations. Representing unknowns as linear combinations allows compact, elegant modeling.", "---", "### Conclusion", "Solving matrix equations like\n[\n\begin{pmatrix} a(1) + b(3) & a(2) + b(4) \ c(1) + d(3) & c(2) + d(4) \end{pmatrix} = \begin{pmatrix} 5 & 6 \ 7 & 8 \end{pmatrix}\n]\nis reduced to solving standard linear systems. By matching corresponding entries, we decouple equations and solve for each variable uniquely.", "Understanding this structure empowers you to tackle more complex systems and reinforces foundational linear algebra skills vital for advanced study and real-world problem-solving.", "---", "Keywords: matrix equation, linear algebra, linear systems, solving equations, vector expressions, matrix correspondence, mathematical modeling"]