A system of two linear equations in two variables is shown below. 4x+2y=8 y=3x-6 Joakim claims that there exists a matrix [} a&b c&d ] represents the system of linear equations. For the case where the claim is true, what must be the matrix [} a&b c&d ] A [} 4&2 1&3 ] B. [} 4&2 3&1 ] C. [} 4&2 1&-3 ] D. [} 4&2 -3&1 ]

To determine which matrix represents the system of linear equations, we need to multiply the matrix by the vector $[\begin{matrix} x\\ y\end{matrix} ]$ and see if it results in the vector $[\begin{matrix} 8\\ -6\end{matrix} ]$.<br /><br />Let's start by multiplying the matrix $[\begin{matrix} a&b\\ c&d\end{matrix} ]$ by the vector $[\begin{matrix} x\\ y\end{matrix} ]$:<br /><br />$[\begin{matrix} a&b\\ c&d\end{matrix} ][\begin{matrix} x\\ y\end{matrix} ] = [\begin{matrix} ax+by\\ cx+dy\end{matrix} ]$<br /><br />We want this to be equal to $[\begin{matrix} 8\\ -6\end{matrix} ]$, so we have the following system of equations:<br /><br />$ax+by = 8$<br />$cx+dy = -6$<br /><br />Now let's compare these equations to the original system of linear equations:<br /><br />$4x+2y = 8$<br />$y = 3x-6$<br /><br />We can see that the first equation matches the first equation in the original system, so we need to find the values of $a$, $b$, $c$, and $d$ that satisfy the second equation.<br /><br />Substituting $y = 3x-6$ into the second equation, we get:<br /><br />$cx + d(3x-6) = -6$<br /><br />Expanding and simplifying, we have:<br /><br />$(c+3d)x - 6d = -6$<br /><br />Comparing this to the original second equation, we can see that $c+3d = 0$ and $-6d = -6$. Solving for $d$, we get $d = 1$. Substituting this back into $c+3d = 0$, we find $c = -3$.<br /><br />So the matrix that represents the system of linear equations is $[\begin{matrix} 4&2\\ -3&1\end{matrix} ]$, which corresponds to option D.