{:[2y+16=10 x],[4x-Ly=8-x]:} In the system of equations, L is a constant. For what value of L does the system of linear equations have infinitely many solutions?

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{:[2y+16=10 x],[4x-Ly=8-x]:} In the system of equations,  L is a constant. For what value of  L does the system of linear equations have infinitely many solutions?

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Write Equations: First, let's write down the system of equations: $$ \begin{cases}  2y + 16 = 10x \ 4x - Ly = 8 - x  \end{cases} $$ To have infinitely many solutions, the two equations must be dependent, meaning one is a multiple of the other.Solve for y: Let's solve the first equation for $ y $ to get it in the form $ y = mx + b $, where $ m $ is the slope and $ b $ is the y-intercept. $$ 2y = 10x - 16 $$ $$ y = 5x - 8 $$Express Second Equation: Now, let's express the second equation in terms of $ y $ as well: $$ 4x - Ly = 8 - x $$ $$ Ly = 4x - 8 + x $$ $$ Ly = 5x - 8 $$Check Coefficients: For the system to have infinitely many solutions, the equations must represent the same line. Therefore, the coefficients of $ x $ and the constants must be the same in both equations. From the first equation, we have the coefficient of $ x $ as 5 and the constant as -8. From the second equation, we have the coefficient of $ x $ as 5 (which matches) and the constant as -8 (which also matches). The coefficient of $ y $ in the second equation must be 1 for the equations to be the same, since in the first equation, the coefficient of $ y $ is implicitly 1. $$ L = 1 $$

$\begin{array}{c} 2 y+16=10 x \\ 4 x-L y=8-x \end{array}$ $\newline$ In the system of equations, $L$ is a constant. For what value of $L$ does the system of linear equations have infinitely many solutions?

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2y+16=10x4x−Ly=8−x \begin{array}{c} 2 y+16=10 x \\ 4 x-L y=8-x \end{array} 2y+16=10x4x−Ly=8−x​\newlineIn the system of equations, L L L is a constant. For what value of L L L does the system of linear equations have infinitely many solutions?

$\begin{array}{c} 2 y+16=10 x \\ 4 x-L y=8-x \end{array}$ $\newline$ In the system of equations, $L$ is a constant. For what value of $L$ does the system of linear equations have infinitely many solutions?