Question

Solve the following equations by the substitution method :

$$\begin{gathered} \left(viii\right)\frac{1}{2}(9x+10y)=23 \\ \frac{5x}{4}-2y=3 \end{gathered}$$

Answer 1

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

$\frac{1}{2}(9x+10y)=23$

Multiplying the equation throughout by 2:

$$\begin{gathered} 9x+10y=46 \\ 9x=46-10y \\ x=\frac{46-10y}{9} \end{gathered}$$

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

$\frac{5x}{4}-2y=3$

Multiplying the equation throughout by 4:

$$\begin{gathered} 5x-8y=12 \\ 5\left(\frac{46-10y}{9}\right)-8y=12 \end{gathered}$$

Multiplying the equation throughout by 9:

$$\begin{gathered} 5(46-10y)-72y=108 \\ 230-50y-72y=108 \\ -122y=108-230 \\ -122y=-122 \\ y=\frac{-122}{-122} \\ y=1 \end{gathered}$$

Step 3: Substitute the values of y obtained in $x=\frac{46-10y}{9}$ to get the values of x:

$$\begin{gathered} x=\frac{46-10y}{9} \\ x=\frac{46-(10\times 1)}{9} \\ x=\frac{46-10}{9} \\ x=\frac{36}{9} \\ x=4 \end{gathered}$$

Hence, the value of $x=4andy=1$.