Solve {l}{6+y/5-2+x=x+y-2}{3x+2/4+5y+8/3=5x+7y}

$$\left. \begin{array} { l } { \frac { 6 + y } { 5 } - 2 + x = x + y - 2 } \\ { \frac { 3 x + 2 } { 4 } + \frac { 5 y + 8 } { 3 } = 5 x + 7 y } \end{array} \right.$$

Solve for y, x

$$x = -\frac{58}{51} = -1\frac{7}{51} \approx -1.137254902$$

$$y = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

Solution Steps

Consider the first equation. Multiply both sides of the equation by 5.

$$6+y-10+5x=5x+5y-10$$

Subtract 10 from 6 to get -4.

$$-4+y+5x=5x+5y-10$$

Subtract 5x from both sides.

$$-4+y+5x-5x=5y-10$$

Combine 5x and -5x to get 0.

$$-4+y=5y-10$$

Subtract 5y from both sides.

$$-4+y-5y=-10$$

Combine y and -5y to get -4y.

$$-4-4y=-10$$

Add 4 to both sides.

$$-4y=-10+4$$

Add -10 and 4 to get -6.

$$-4y=-6$$

Divide both sides by -4.

$$y=\frac{-6}{-4}$$

Reduce the fraction \frac{-6}{-4} to lowest terms by extracting and canceling out -2.

$$y=\frac{3}{2}$$

Consider the second equation. Insert the known values of variables into the equation.

$$\frac{3x+2}{4}+\frac{5\times \frac{3}{2}+8}{3}=5x+7\times \frac{3}{2}$$

Multiply both sides of the equation by 12, the least common multiple of 4,3,2.

$$3\left(3x+2\right)+4\left(5\times \frac{3}{2}+8\right)=60x+84\times \frac{3}{2}$$

Use the distributive property to multiply 3 by 3x+2.

$$9x+6+4\left(5\times \frac{3}{2}+8\right)=60x+84\times \frac{3}{2}$$

Multiply 5 and \frac{3}{2} to get \frac{15}{2}.

$$9x+6+4\left(\frac{15}{2}+8\right)=60x+84\times \frac{3}{2}$$

Add \frac{15}{2} and 8 to get \frac{31}{2}.

$$9x+6+4\times \frac{31}{2}=60x+84\times \frac{3}{2}$$

Multiply 4 and \frac{31}{2} to get 62.

$$9x+6+62=60x+84\times \frac{3}{2}$$

Add 6 and 62 to get 68.

$$9x+68=60x+84\times \frac{3}{2}$$

Multiply 84 and \frac{3}{2} to get 126.

$$9x+68=60x+126$$

Subtract 60x from both sides.

$$9x+68-60x=126$$

Combine 9x and -60x to get -51x.

$$-51x+68=126$$

Subtract 68 from both sides.

$$-51x=126-68$$

Subtract 68 from 126 to get 58.

$$-51x=58$$

Divide both sides by -51.

$$x=-\frac{58}{51}$$

The system is now solved.

$$y=\frac{3}{2} x=-\frac{58}{51}$$