{:[x=(1)/(3)(y+4)],[4x-3y=3(a-x)-2x]:} In the system of equations, a is a constant. For what value of a does the system of linear equations have infinitely many solutions?

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{:[x=(1)/(3)(y+4)],[4x-3y=3(a-x)-2x]:} In the system of equations,  a is a constant. For what value of  a does the system of linear equations have infinitely many solutions?

Bytelearn · Accepted Answer

Write down the system of equations: First, let's write down the system of equations: 1. $ x = \frac{1}{3}(y + 4) $ 2. $ 4x - 3y = 3(a - x) - 2x $ For a system of equations to have infinitely many solutions, both equations must represent the same line. This means that the equations must be proportional or one must be a multiple of the other.Express the first equation in terms of y: Let's express the first equation in terms of y to compare it with the second equation: $ x = \frac{1}{3}(y + 4) $ Multiply both sides by 3 to get rid of the fraction: $ 3x = y + 4 $ Now, let's isolate y: $ y = 3x - 4 $Rewrite the second equation to isolate y: Next, we will rewrite the second equation to isolate y as well: $ 4x - 3y = 3(a - x) - 2x $ First, distribute the 3 on the right side: $ 4x - 3y = 3a - 3x - 2x $ Combine like terms on the right side: $ 4x - 3y = 3a - 5x $ Now, let's isolate y by adding 3y to both sides and adding 5x to both sides: $ 4x + 5x = 3y + 3a $ $ 9x = 3y + 3a $ Divide both sides by 3 to solve for y: $ 3x = y + a $ Now, let's isolate y: $ y = 3x - a $Compare the two expressions for y: Now we have two expressions for y: From the first equation: $ y = 3x - 4 $ From the second equation: $ y = 3x - a $ For the system to have infinitely many solutions, these two expressions must be identical, which means that the constants on the right side of each equation must be equal: $ -4 = -a $Solve for a: Solve for a: $ a = 4 $

$\begin{array}{c} x=\frac{1}{3}(y+4) \\ 4 x-3 y=3(a-x)-2 x \end{array}$ $\newline$ In the system of equations, $a$ is a constant. For what value of $a$ does the system of linear equations have infinitely many solutions?

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$\begin{array}{c} x=\frac{1}{3}(y+4) \\ 4 x-3 y=3(a-x)-2 x \end{array}$ $\newline$ In the system of equations, $a$ is a constant. For what value of $a$ does the system of linear equations have infinitely many solutions?