Question

Solve the following equations by the substitution method :

$$\begin{gathered} \left(vi\right)12x-16y=20 \\ 8x+6y=30 \end{gathered}$$

Answer 1

Step 1: Express one variable (say x) in terms of the other variable (say y) from one of the given equations:

$$\begin{gathered} 12x-16y=20 \\ 12x=20+16y \\ x=\frac{20+16y}{12} \end{gathered}$$

Step 2: Substitute this value of x in the other equations to get a linear equation in y which can be solved:

$$\begin{gathered} 8x+6y=30 \\ 8\left(\frac{20+16y}{12}\right)+6y=30 \end{gathered}$$

Multiplying the equation throughout by 12:

$$\begin{gathered} 8(20+16y)+72y=360 \\ 160+128y+72y=360 \\ 200y=360-160 \\ 200y=200 \\ y=\frac{200}{200} \\ y=1 \end{gathered}$$

Step 3: Substitute the values of y obtained in $x=\frac{20+16y}{12}$ to get the values of x:

$$\begin{gathered} x=\frac{20+16y}{12} \\ x=\frac{20+(16\times 1)}{12} \\ x=\frac{20+16}{12} \\ x=\frac{36}{12} \\ x=3 \end{gathered}$$

Hence, the value of $x=3andy=1$.