Question

Solve the following system of equations by the elimination method and the substitution method:

$$\begin{gathered} 2x-4y=1 \\ 5x+3y=4 \end{gathered}$$

Answer 1

Elimination method

Step(1): Making coefficient of variable$(xory)$equal

Multiply the given equation needed by a suitable number so that the coefficient of one of the variables says$(xory)$ become equal.

Multiply by $5$ in $2x-4y=1$ and multiply by $2$ in $5x+3y=4$ which make $x$ coefficient equal.

$$\begin{gathered} 10x-20y=5 \\ 10x+6y=8 \end{gathered}$$

Step (2): Eliminating the variable $x$

Subtract the equations obtained in step-$\left(1\right)$ to eliminate one variable. The resulting equation is a linear equation in one variable..

$$\begin{gathered} -20y-6y=5-8 \\ \Rightarrow -26y=-3 \end{gathered}$$

$\begin{array}{rcl}& \Rightarrow & 26y=3\\ \Rightarrow y& =& \frac{3}{26}\end{array}$

Step (3): Obtain value of other variable

Substitute this value of the variable in either of the original equations, to get the value of the other variable.

$$\begin{gathered} 10x+6y=8 \\ \Rightarrow 10x+(6\times \frac{3}{26})=8 \\ \Rightarrow 10x+\frac{18}{26}=8 \end{gathered}$$

multiplying the equation throughout by $26$

$\begin{array}{rcl}260x+18& =& 208\\ \Rightarrow 260x& =& 208-18\\ \Rightarrow 260x& =& 190\\ \Rightarrow x& =& \frac{190}{260}\\ \Rightarrow x& =& \frac{19}{26}\end{array}$

Hence $x=\frac{19}{26}\&y=\frac{3}{26}$

Substitution Method

Step(1):Express one variable in terms of other

Express one variable $(sayx)$ in terms of other variable $(sayy)$ from one of the given equations.

$$\begin{gathered} 2x-4y=1 \\ \Rightarrow 2x=1+4y \\ \Rightarrow x=\frac{1+4y}{2} \end{gathered}$$

Step (2): Use the second equation

Substitute this value of x in the other equations to get a linear equations in $y$ which can be solved.

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

multiplying the equation throughout by 2

$\begin{array}{rcl}5(1+4y)+6y& =& 8\\ \Rightarrow 5+20y+6y& =& 8\\ \Rightarrow 26y& =& 8-5\\ \Rightarrow y& =& \frac{3}{26}\end{array}$

Step (3) : Finding the other variable.

Substitute the values of $y$ obtained in step-$\left(2\right)$ above in the equations used in step- $\left(1\right)$to get the values of $x$.

$$\begin{gathered} x=\frac{1+4y}{2} \\ \Rightarrow x=\frac{1+{\displaystyle \frac{4\times 3}{26}}}{2} \\ \Rightarrow x=\frac{26+12}{2\times 26} \\ \Rightarrow x=\frac{19}{26} \end{gathered}$$

Hence,by both mehod answer is same and $x=\frac{19}{26}\&y=\frac{3}{26}$