Solve {l}{x=y-2}{y=log(e^3/120)}

$$\left. \begin{array} { l } { x = y - 2 } \\ { y = \log ( \frac { e ^ { 3 } } { 120 } ) } \end{array} \right.$$

Solve for x, y

$$x=-\log(e)\left(\ln(12000)-3\right)\approx -2.7762978$$

$$y=\log(e)\left(3-\ln(120)\right)\approx -0.7762978$$

Steps Using Substitution

Consider the first equation. Subtract y from both sides.

$$x-y=-2$$

To solve a pair of equations using substitution, first solve one of the equations for one of the variables. Then substitute the result for that variable in the other equation.

$$y=\log_{10}\left(\frac{e^{3}}{120}\right),-y+x=-2$$

Pick one of the two equations which is more simple to solve for y by isolating y on the left hand side of the equal sign.

$$y=\log_{10}\left(\frac{e^{3}}{120}\right)$$

Divide both sides by 1.

$$y=\log(e)\left(3-\ln(120)\right)$$

Substitute \left(-\ln(120)+3\right)\log(e) for y in the other equation, -y+x=-2.

$$-\log(e)\left(3-\ln(120)\right)+x=-2$$

Add \left(-\ln(120)+3\right)\log(e) to both sides of the equation.

$$x=\log(e)\left(\ln(\frac{1}{12000})+3\right)$$

The system is now solved.

$$y=\log(e)\left(3-\ln(120)\right),x=\log(e)\left(3-\ln(12000)\right)$$