Question

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

$\begin{array}{rcl}y-2x& =& 3\\ y-4x& =& 2\end{array}$

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.

Here, $y$ has same coefficient

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

Step (2): Eliminating the variable $y$

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

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

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

$\Rightarrow x=\frac{1}{2}$

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.

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

Hence $x=\frac{1}{2}\&y=4$

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.

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

$$\begin{gathered} y-2x=3 \\ \Rightarrow y-3=2x \\ \Rightarrow x=\frac{y-3}{2} \end{gathered}$$

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.

$$\begin{gathered} y-4x=2 \\ y-4\left(\frac{y-3}{2}\right)=2 \\ \Rightarrow y-2(y-3)=2 \\ \Rightarrow y-2y+6=2 \\ \Rightarrow -y=2-6 \\ \Rightarrow y=4 \end{gathered}$$

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 $x$.

$$\begin{gathered} x=\frac{y-3}{2} \\ \Rightarrow x=\frac{4-3}{2} \\ \Rightarrow x=\frac{1}{2} \end{gathered}$$

Hence,by both mehod answer is same and $x=\frac{1}{2}\&y=4$