Question

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

$\begin{array}{rcl}x& =& 2y+5\\ 7y& =& 2x-7\end{array}$

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 $2$ in $x=2y+5$

$$\begin{gathered} 2x=4y+10 \\ 7y=2x-7 \end{gathered}$$

i.e. $-2x=-7y-7......................................................\left(1\right)$

Step (2):Eliminating the variable $x$

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

$0=-3y+3................................................\left(2\right)$.

$$\begin{gathered} -3=-3y \\ \Rightarrow y=\frac{3}{3} \\ \Rightarrow y=1 \end{gathered}$$

Step (3): Obtain value of other variable

Substitute this value of $y=1$ to the variable in either of the original equations, to get the value of the other variable.

$$\begin{gathered} x=2y+5 \\ \Rightarrow x=(2\times 1)+5 \\ \Rightarrow x=2+5 \\ \Rightarrow x=7 \end{gathered}$$

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.

$x=2y+5........................................\left(1\right)$

Step (2): Use the second equation

Substitute this value of $x=2y+5$ in the other equations to get a linear equations in $y$ which can be solved.

$$\begin{gathered} 7y=2x-7....................................................\left(2\right) \\ \Rightarrow 7y=2(2y+5)-7 \\ \Rightarrow 7y=4y+10-7 \\ \Rightarrow 7y-4y=3 \\ \Rightarrow 3y=3 \\ \Rightarrow y=1 \end{gathered}$$

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 $x$.

$$\begin{gathered} x=2y+5 \\ \Rightarrow x=(2\times 1)+5 \\ \Rightarrow x=2+5 \\ \Rightarrow x=7 \end{gathered}$$

Hence,by both mehod answer is same and $x=7$, $y=1$