Solve {l}{x+y/3x-y/4=5}{x+y/3+x-y/4=11}

$$\left\{ \begin{array} { l } { \frac { x + y } { 3 } \frac { x - y } { 4 } = 5 } \\ { \frac { x + y } { 3 } + \frac { x - y } { 4 } = 11 } \end{array} \right.$$

Solve for x, y

$$x=\frac{\sqrt{101}+77}{4}\approx 21.762468905\text{, }y=\frac{-7\sqrt{101}-11}{4}\approx -20.337282337$$

$$x=\frac{77-\sqrt{101}}{4}\approx 16.737531095\text{, }y=\frac{7\sqrt{101}-11}{4}\approx 14.837282337$$

Steps Using Substitution and Quadratic Formula

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

$$\left(x+y\right)\left(x-y\right)=60$$

Consider \left(x+y\right)\left(x-y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$x^{2}-y^{2}=60$$

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

$$4\left(x+y\right)+3\left(x-y\right)=132$$

Use the distributive property to multiply 4 by x+y.

$$4x+4y+3\left(x-y\right)=132$$

Use the distributive property to multiply 3 by x-y.

$$4x+4y+3x-3y=132$$

Combine 4x and 3x to get 7x.

$$7x+4y-3y=132$$

Combine 4y and -3y to get y.

$$7x+y=132$$

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.

$$7x+y=132,-y^{2}+x^{2}=60$$

Solve 7x+y=132 for x by isolating x on the left hand side of the equal sign.

$$7x+y=132$$

Subtract y from both sides of the equation.

$$7x=-y+132$$

Divide both sides by 7.

$$x=-\frac{1}{7}y+\frac{132}{7}$$

Substitute -\frac{1}{7}y+\frac{132}{7} for x in the other equation, -y^{2}+x^{2}=60.

$$-y^{2}+\left(-\frac{1}{7}y+\frac{132}{7}\right)^{2}=60$$

Square -\frac{1}{7}y+\frac{132}{7}.

$$-y^{2}+\frac{1}{49}y^{2}-\frac{264}{49}y+\frac{17424}{49}=60$$

Add -y^{2} to \frac{1}{49}y^{2}.

$$-\frac{48}{49}y^{2}-\frac{264}{49}y+\frac{17424}{49}=60$$

Subtract 60 from both sides of the equation.

$$-\frac{48}{49}y^{2}-\frac{264}{49}y+\frac{14484}{49}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1+1\left(-\frac{1}{7}\right)^{2} for a, 1\times \frac{132}{7}\left(-\frac{1}{7}\right)\times 2 for b, and \frac{14484}{49} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-\frac{264}{49}\right)±\sqrt{\left(-\frac{264}{49}\right)^{2}-4\left(-\frac{48}{49}\right)\times \frac{14484}{49}}}{2\left(-\frac{48}{49}\right)}$$

Square 1\times \frac{132}{7}\left(-\frac{1}{7}\right)\times 2.

$$y=\frac{-\left(-\frac{264}{49}\right)±\sqrt{\frac{69696}{2401}-4\left(-\frac{48}{49}\right)\times \frac{14484}{49}}}{2\left(-\frac{48}{49}\right)}$$

Multiply -4 times -1+1\left(-\frac{1}{7}\right)^{2}.

$$y=\frac{-\left(-\frac{264}{49}\right)±\sqrt{\frac{69696}{2401}+\frac{192}{49}\times \frac{14484}{49}}}{2\left(-\frac{48}{49}\right)}$$

Multiply \frac{192}{49} times \frac{14484}{49} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\left(-\frac{264}{49}\right)±\sqrt{\frac{69696+2780928}{2401}}}{2\left(-\frac{48}{49}\right)}$$

Add \frac{69696}{2401} to \frac{2780928}{2401} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\left(-\frac{264}{49}\right)±\sqrt{\frac{58176}{49}}}{2\left(-\frac{48}{49}\right)}$$

Take the square root of \frac{58176}{49}.

$$y=\frac{-\left(-\frac{264}{49}\right)±\frac{24\sqrt{101}}{7}}{2\left(-\frac{48}{49}\right)}$$

The opposite of 1\times \frac{132}{7}\left(-\frac{1}{7}\right)\times 2 is \frac{264}{49}.

$$y=\frac{\frac{264}{49}±\frac{24\sqrt{101}}{7}}{2\left(-\frac{48}{49}\right)}$$

Multiply 2 times -1+1\left(-\frac{1}{7}\right)^{2}.

$$y=\frac{\frac{264}{49}±\frac{24\sqrt{101}}{7}}{-\frac{96}{49}}$$

Now solve the equation y=\frac{\frac{264}{49}±\frac{24\sqrt{101}}{7}}{-\frac{96}{49}} when ± is plus. Add \frac{264}{49} to \frac{24\sqrt{101}}{7}.

$$y=\frac{\frac{24\sqrt{101}}{7}+\frac{264}{49}}{-\frac{96}{49}}$$

Divide \frac{264}{49}+\frac{24\sqrt{101}}{7} by -\frac{96}{49} by multiplying \frac{264}{49}+\frac{24\sqrt{101}}{7} by the reciprocal of -\frac{96}{49}.

$$y=\frac{-7\sqrt{101}-11}{4}$$

Now solve the equation y=\frac{\frac{264}{49}±\frac{24\sqrt{101}}{7}}{-\frac{96}{49}} when ± is minus. Subtract \frac{24\sqrt{101}}{7} from \frac{264}{49}.

$$y=\frac{-\frac{24\sqrt{101}}{7}+\frac{264}{49}}{-\frac{96}{49}}$$

Divide \frac{264}{49}-\frac{24\sqrt{101}}{7} by -\frac{96}{49} by multiplying \frac{264}{49}-\frac{24\sqrt{101}}{7} by the reciprocal of -\frac{96}{49}.

$$y=\frac{7\sqrt{101}-11}{4}$$

There are two solutions for y: \frac{-11-7\sqrt{101}}{4} and \frac{-11+7\sqrt{101}}{4}. Substitute \frac{-11-7\sqrt{101}}{4} for y in the equation x=-\frac{1}{7}y+\frac{132}{7} to find the corresponding solution for x that satisfies both equations.

$$x=-\frac{1}{7}\times \frac{-7\sqrt{101}-11}{4}+\frac{132}{7}$$

Multiply -\frac{1}{7} times \frac{-11-7\sqrt{101}}{4}.

$$x=-\frac{-7\sqrt{101}-11}{4\times 7}+\frac{132}{7}$$

Now substitute \frac{-11+7\sqrt{101}}{4} for y in the equation x=-\frac{1}{7}y+\frac{132}{7} and solve to find the corresponding solution for x that satisfies both equations.

$$x=-\frac{1}{7}\times \frac{7\sqrt{101}-11}{4}+\frac{132}{7}$$

Multiply -\frac{1}{7} times \frac{-11+7\sqrt{101}}{4}.

$$x=-\frac{7\sqrt{101}-11}{4\times 7}+\frac{132}{7}$$

The system is now solved.

$$x=-\frac{-7\sqrt{101}-11}{4\times 7}+\frac{132}{7},y=\frac{-7\sqrt{101}-11}{4}\text{ or }x=-\frac{7\sqrt{101}-11}{4\times 7}+\frac{132}{7},y=\frac{7\sqrt{101}-11}{4}$$