Solving Systems of Equations: Substitution of Fractions in 2x-y/2 + -3y+x/5 = 7

To solve the system of equations using substitution:

Step 1: Simplify each equation.

First equation:

$\frac{2x - y}{2} + \frac{-3y + x}{5} = 7$

Multiply by 10 to eliminate denominators:

$(2x - y) \times 5 + (-3y + x) \times 2 = 70$ $10x - 5y - 6y + 2x = 70$ $12x - 11y = 70\quad\text{(1)}$

Second equation:

$\frac{-y - x}{8} - \frac{5y + x}{6} = 4$

Multiply by 24 to eliminate denominators:

$( -y - x) \times 3 - (5y + x) \times 4 = 96$ $-3y - 3x - 20y - 4x = 96$ $-7x - 23y = 96\quad\text{(2)}$

Step 2: Solve the first equation for $x$:

$12x = 70 + 11y$ $x = \frac{70 + 11y}{12}\quad\text{(3)}$

Step 3: Substitute Equation (3) into Equation (2):

$-7\left(\frac{70 + 11y}{12}\right) - 23y = 96$

Clear while multiplying by 12:

$-7(70 + 11y) - 276y = 1152$ $-490 - 77y - 276y = 1152$ $-353y = 1642$ $y = -4.65$

Step 4: Substitute $y = -4.65$ back into Equation (3) to find $x$:

$x = \frac{70 + 11(-4.65)}{12}$ $x = \frac{70 - 51.15}{12}$ $x = \frac{18.85}{12}$ $x = 1.57$

Therefore, the solution to the problem is $x = 1.57$, $y = -4.65$.