Chapter 11.1, Problem 9AYU

$\{\begin{array}{l}3x-\text{ 4}y=\text{ 4 }\\ \frac{1}{2}x-\text{ 3}y=-\frac{1}{2}\end{array}$

$x=\text{ 2},y=\frac{1}{2};\left(2,\frac{1}{2}\right)$

To determine

The solution of the given system of equations.

Answer to Problem 9AYU

$\left(x,y\right)=\left(2,\frac{1}{2}\right)$ is the solution.

Explanation of Solution

Given:

$3x-4y=\text{ 4}$

$\frac{1}{2}x-3y=-\frac{1}{2}$

$x=2,y=\frac{1}{2}$

Calculation:

The system of equations is,

$3x-4y=\text{ 4}$

$\frac{1}{2}x-3y=-\frac{1}{2}$

From the second equation we get,

$\frac{1}{2}x=3y-\frac{1}{2}$

$x=6y-1$

Substituting this value of $x$ in the first equation we get,

$3\left(6y-1\right)-4y=4$

$18y-3-4y=4$

$14y=4+3$

$14y=7$

$y=\frac{7}{14}$

$y=\frac{1}{2}$

Substituting this value of $y$ in the second equation we get,

$\frac{1}{2}x-3\left(\frac{1}{2}\right)=-\frac{1}{2}$

$x-3=-1$

$x=-1+3$

$x=2$

Thus the solution is $x=2,y=\frac{1}{2}$.

Therefore the given point $\left(2,\frac{1}{2}\right)$ satisfies the given system of equations and hence is the solution.