Chapter 11, Problem 2RE

To determine

To find: the equation of $\left\{\begin{array}{l}3x-4y=4\\ x-3y=\frac{1}{2}\end{array}\right\}$

Answer to Problem 2RE

The equation of the solutions is $\left(2,\frac{1}{2}\right)$ .

Explanation of Solution

Given information:

The given equation is $\left\{\begin{array}{l}3x-4y=4\\ x-3y=\frac{1}{2}\end{array}\right\}$

Calculation:

Calculate the equation,

Multiply both sides of the equation $x-3y=\frac{1}{2}$ by $2$ so as to make the coefficients of $y$ negative of one another $2x-6y=1$ .

So, an equivalent system of equation is $\left\{\begin{array}{l}3x-4y=4\\ 2x-6y=1\end{array}\right\}$

In order to eliminate $y$ , multiply the first equation by 3 and the second equation by -2

  $\begin{array}{c}3\left(3x-4y\right)=3\cdot 4\\ 9x-12y=12\end{array}$

  $\begin{array}{c}-2\left(2x-6y\right)=1\times -2\\ -4x+12y=-2\end{array}$

Add the equation,

  $\begin{array}{c}9x-12y=12\\ -4x+12y=-2\\ 5x=10\end{array}$

Divide both sides of the equation by 5.

  $\begin{array}{c}\frac{5x}{5}=\frac{10}{5}\\ x=2\end{array}$

Substitute 2 for $x$ i9n the equation $2x-6y=1$ and solve for $y$

  $\begin{array}{c}2\times 2-6y=1\\ 4-6y=1\end{array}$

Subtract 2 from both sides of the equation and simplify.

  $\begin{array}{c}4-6y-4=1-4\\ -6y=-3\\ y=\frac{1}{2}\end{array}$

Therefore, the solution expressed in ordered form is $\left(2,\frac{1}{2}\right)$

In order to check the solution , substitute 2 for $x$ and $\frac{1}{2}$ for $y$ in the given system of equations.

Check: $\begin{array}{l}\left\{\begin{array}{l}3x-4y=4\\ x-3y=\frac{1}{2}\end{array}\right\}\\ 3\times 2-4\times \frac{1}{2}=4\\ 25-3\times \frac{1}{2}=\frac{1}{2}\end{array}$

  $$ Therefore, the solution is checks.