Question

Solve the system by the method of substitution.

{3x2y=36x+4y=6\left\{\begin{array}{rr} 3 x-2 y= & 3 \\ -6 x+4 y= & -6 \end{array}\right.

Solution

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{3x2y=3 16x+4y=6 2From the equation 1 y=3x32Substitute y=3x32 into the equation 26x+4y=66x+4(3x32)=66x+2(3x3)=6Distributive property6x+6x6=6Solve for x6x+6x=6+60=0The equation 0=0 is true for any value of x.Therefore,The system of linear equations is dependent and has infinitely many solutions\begin{gathered} \left\{ {\begin{array}{c} {3x - 2y = 3{\text{ }}\boxed1} \\ { - 6x + 4y = - 6{\text{ }}\boxed2} \end{array}} \right. \\ {\text{From the equation }}\boxed{\text{1}}{\text{ }} \\ y = \frac{{3x - 3}}{2} \\ {\text{Substitute }}y = \frac{{3x - 3}}{2}{\text{ into the equation }}\boxed2 \\ \underbrace { - 6x + 4y = - 6}_ \Downarrow \\ - 6x + 4\left( {\frac{{3x - 3}}{2}} \right) = - 6 \\ - 6x + 2\left( {3x - 3} \right) = - 6 \\ {\text{Distributive property}} \\ - 6x + 6x - 6 = - 6 \\ {\text{Solve for }}x \\ - 6x + 6x = - 6 + 6 \\ 0 = 0 \\ {\text{The equation 0}} = 0{\text{ is true for any value of }}x. \\ {\text{Therefore}}{\text{,}} \\ {\text{The system of linear equations is dependent and has infinitely many solutions}} \\ \end{gathered}

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