Question

9) solve using substitution method and check your answer:4x - 3y + 2z = 16- 4y - Z = 7= 146x - y

Answer 1

Given the system of equations, solve the third equation for y, as shown below

`\begin{gathered} 6x-y=14 \\ \Rightarrow y=6x-14 \end{gathered}`

And, solve for z in the second equation,

`\begin{gathered} -4y-z=7 \\ \Rightarrow z=-4y-7 \\ \Rightarrow z=-4(6x-14)-7=-24x+49 \end{gathered}`

Thus, substitute the values of y and z in terms of x into the first equation, as shown below

`\begin{gathered} \Rightarrow4x-3y+2z=4x-3(6x-14)+2(-24x+49)=4x-18x+42-48x+98 \\ \Rightarrow-62x+140=16 \\ \Rightarrow-62x=-124 \\ \Rightarrow x=2 \end{gathered}`

Then, solving for y and z given x=2,

`\begin{gathered} x=2 \\ \Rightarrow y=6*2-14=-2 \\ and \\ z=-24*2+49=-48+49=1 \end{gathered}`

Therefore, the solution to the system of equations is x=2, y=-2, z=1

To verify the solutions, substitute the values we found into the three equations of the system, as shown below

`\begin{gathered} x=2,y=-2,z=1 \\ \Rightarrow4x-3y+2z=4*2-3*(-2)+2*1=8+6+2=16\rightarrow correct \\ \Rightarrow-4y-z=-4*(-2)-1(1)=8-1=7\rightarrow correct \\ \Rightarrow6x-y=6*2-1(-2)=12+2=14\rightarrow correct \end{gathered}`