Question

Systems of equations: solve using substitution method
$$\begin{gathered} y-6x=-15 \\ z-3x=2 \\ x+2y+3z=86 \end{gathered}$$

Answer 1

Substitution method:

One of the equations is transformed to give the relation of one variable with respect to other.

Then this variable is substituted by the relation in the other equation so that the second equation has one variable only.

$$\begin{gathered} y-6x=-15\to \left(1\right) \\ z-3x=2\to \left(2\right) \\ x+2y+3z=86\to \left(3\right) \end{gathered}$$

Step-1: Solve equation $1$ and $2$ in terms of $x$:

Equation $1$

$$\begin{gathered} y-6x=-15 \\ \Rightarrow y=6x-15 \end{gathered}$$

Equation $2$

$$\begin{gathered} z-3x=2 \\ \Rightarrow z=2+3x \end{gathered}$$

Step-2: Substituting the value of $y\text{and}z$ in equation $3$:

$$\begin{gathered} x+2y+3z=86 \\ \Rightarrow x+2\times \left(6x-15\right)+3\left(2+3x\right)=86 \\ \Rightarrow x+12x-30+6+9x=86 \\ \Rightarrow 22x=86+24 \\ \Rightarrow 22x=110 \\ \Rightarrow x=\frac{110}{22} \\ \Rightarrow x=5 \end{gathered}$$

Step-3: Substituting the value $x=5$ in $y\text{and}z$:

Substitute $x=5$ in $y=6x-15$.

$$\begin{gathered} y=6\times 5-15 \\ =30-15 \\ =15 \end{gathered}$$

Substitute $x=5$ in $z=2+3x$.

$$\begin{gathered} z=2+\left(3\times 5\right) \\ =2+15 \\ =17 \end{gathered}$$

Hence, the value of $x=5$,$y=15$ and $z=17$.