Chapter 7.2, Problem 65E

To determine

Whether the method of substitution is more easier than the method of elimination.

Explanation of Solution

Given:

Given system of equation is

  $\begin{array}{l}3x+2y=4\\ 5x-2y=12\end{array}$

Calculation:

Consider the given system of equation and solve the system with the method of substitution.

  $\begin{array}{l}3x+2y=4\\ 5x-2y=12\end{array}$

Solve the first equation for $y$ , therefore,

  $\begin{array}{c}3x+2y=4\\ 2y=4-3x\\ y=\frac{4-3x}{2}\end{array}$

Substitute the expression $y=\frac{4-3x}{2}$ obtained in the previous step, in the second equation.

Use the distributive property of addition and combine the like term to obtain single variable equation in terms of $x$ .

Therefore,

  $\begin{array}{c}5x-2y=12\\ 5x-2\left(\frac{4-3x}{2}\right)=12\\ 5x-4+3x=12\\ \left(5x+3x\right)+\left(-4-12\right)=0\\ 8x-16=0\end{array}$

Now solve the resulting single variable equation for $x$

  $\begin{array}{c}8x-16=0\\ 8x=16\\ x=2\end{array}$

Back- substitute $x=2$ into the equation $y=\frac{4-3x}{2}$ and solve for $y$ , therefore,

  $\begin{array}{c}y=\frac{4-3x}{2}\\ y=\frac{4-\left(3\times 2\right)}{2}\\ y=\frac{4-6}{2}\\ y=\frac{-2}{2}\end{array}$

Or,

  $y=-1$

Hence, the solution of the given system of equations is the ordered pair $\left(2,-1\right)$ .

To solve this system of equations using the method of elimination is much easier than solving the system using the method of substitution. As the coefficient of $y$ in the both equations is same but with a different sign, so one can directly add the two equations of the system and find the value of $x$ .And then by back substituting the value of $x$ in any one of the equation of the system we can easily find $y$ .