Solve {l}{y/x+55=1/sqrt{3}}{x/y+21=0.64}

$$\left. \begin{array} { l } { \frac { y } { x + 55 } = \frac { 1 } { \sqrt { 3 } } } \\ { \frac { x } { y + 21 } = 0.64 } \end{array} \right.$$

Solve for y, x

$$x = \frac{27376 \sqrt{3} + 39280}{1619} \approx 53.549489134$$

$$y = \frac{42775 \sqrt{3} + 27376}{1619} \approx 62.671076772$$

Steps Using Substitution

Consider the first equation. Variable x cannot be equal to -55 since division by zero is not defined. Multiply both sides of the equation by x+55.

$$y=\frac{1}{3}\left(x+55\right)\times 3^{\frac{1}{2}}$$

Use the distributive property to multiply \frac{1}{3} by x+55.

$$y=\left(\frac{1}{3}x+\frac{55}{3}\right)\times 3^{\frac{1}{2}}$$

Use the distributive property to multiply \frac{1}{3}x+\frac{55}{3} by 3^{\frac{1}{2}}.

$$y=\frac{1}{3}x\times 3^{\frac{1}{2}}+\frac{55}{3}\times 3^{\frac{1}{2}}$$

Subtract \frac{1}{3}x\times 3^{\frac{1}{2}} from both sides.

$$y-\frac{1}{3}x\times 3^{\frac{1}{2}}=\frac{55}{3}\times 3^{\frac{1}{2}}$$

Reorder the terms.

$$y-\frac{1}{3}\sqrt{3}x=\frac{55}{3}\sqrt{3}$$

Consider the second equation. Variable y cannot be equal to -21 since division by zero is not defined. Multiply both sides of the equation by y+21.

$$x=0.64\left(y+21\right)$$

Use the distributive property to multiply 0.64 by y+21.

$$x=0.64y+13.44$$

Subtract 0.64y from both sides.

$$x-0.64y=13.44$$

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.

$$y+\left(-\frac{\sqrt{3}}{3}\right)x=\frac{55\sqrt{3}}{3},-0.64y+x=13.44$$

Choose one of the equations and solve it for y by isolating y on the left hand side of the equal sign.

$$y+\left(-\frac{\sqrt{3}}{3}\right)x=\frac{55\sqrt{3}}{3}$$

Add \frac{\sqrt{3}x}{3} to both sides of the equation.

$$y=\frac{\sqrt{3}}{3}x+\frac{55\sqrt{3}}{3}$$

Substitute \frac{\left(55+x\right)\sqrt{3}}{3} for y in the other equation, -0.64y+x=13.44.

$$-0.64\left(\frac{\sqrt{3}}{3}x+\frac{55\sqrt{3}}{3}\right)+x=13.44$$

Multiply -0.64 times \frac{\left(55+x\right)\sqrt{3}}{3}.

$$\left(-\frac{16\sqrt{3}}{75}\right)x-\frac{176\sqrt{3}}{15}+x=13.44$$

Add -\frac{16\sqrt{3}x}{75} to x.

$$\left(-\frac{16\sqrt{3}}{75}+1\right)x-\frac{176\sqrt{3}}{15}=13.44$$

Add \frac{176\sqrt{3}}{15} to both sides of the equation.

$$\left(-\frac{16\sqrt{3}}{75}+1\right)x=\frac{176\sqrt{3}}{15}+\frac{336}{25}$$

Divide both sides by -\frac{16\sqrt{3}}{75}+1.

$$x=\frac{27376\sqrt{3}+39280}{1619}$$

Substitute \frac{27376\sqrt{3}+39280}{1619} for x in y=\frac{\sqrt{3}}{3}x+\frac{55\sqrt{3}}{3}. Because the resulting equation contains only one variable, you can solve for y directly.

$$y=\frac{\sqrt{3}}{3}\times \frac{27376\sqrt{3}+39280}{1619}+\frac{55\sqrt{3}}{3}$$

Multiply \frac{\sqrt{3}}{3} times \frac{27376\sqrt{3}+39280}{1619}.

$$y=\frac{39280\sqrt{3}}{4857}+\frac{27376}{1619}+\frac{55\sqrt{3}}{3}$$

Add \frac{55\sqrt{3}}{3} to \frac{27376}{1619}+\frac{39280\sqrt{3}}{4857}.

$$y=\frac{42775\sqrt{3}+27376}{1619}$$

The system is now solved.

$$y=\frac{42775\sqrt{3}+27376}{1619},x=\frac{27376\sqrt{3}+39280}{1619}$$

Steps Using Elimination

Consider the first equation. Variable x cannot be equal to -55 since division by zero is not defined. Multiply both sides of the equation by x+55.

$$y=\frac{1}{3}\left(x+55\right)\times 3^{\frac{1}{2}}$$

Use the distributive property to multiply \frac{1}{3} by x+55.

$$y=\left(\frac{1}{3}x+\frac{55}{3}\right)\times 3^{\frac{1}{2}}$$

Use the distributive property to multiply \frac{1}{3}x+\frac{55}{3} by 3^{\frac{1}{2}}.

$$y=\frac{1}{3}x\times 3^{\frac{1}{2}}+\frac{55}{3}\times 3^{\frac{1}{2}}$$

Subtract \frac{1}{3}x\times 3^{\frac{1}{2}} from both sides.

$$y-\frac{1}{3}x\times 3^{\frac{1}{2}}=\frac{55}{3}\times 3^{\frac{1}{2}}$$

Reorder the terms.

$$y-\frac{1}{3}\sqrt{3}x=\frac{55}{3}\sqrt{3}$$

Consider the second equation. Variable y cannot be equal to -21 since division by zero is not defined. Multiply both sides of the equation by y+21.

$$x=0.64\left(y+21\right)$$

Use the distributive property to multiply 0.64 by y+21.

$$x=0.64y+13.44$$

Subtract 0.64y from both sides.

$$x-0.64y=13.44$$

In order to solve by elimination, coefficients of one of the variables must be the same in both equations so that the variable will cancel out when one equation is subtracted from the other.

$$y+\left(-\frac{\sqrt{3}}{3}\right)x=\frac{55\sqrt{3}}{3},-0.64y+x=13.44$$

To make y and -\frac{16y}{25} equal, multiply all terms on each side of the first equation by -0.64 and all terms on each side of the second by 1.

$$-0.64y-0.64\left(-\frac{\sqrt{3}}{3}\right)x=-0.64\times \frac{55\sqrt{3}}{3},-0.64y+x=13.44$$

Simplify.

$$-0.64y+\frac{16\sqrt{3}}{75}x=-\frac{176\sqrt{3}}{15},-0.64y+x=13.44$$

Subtract -0.64y+x=13.44 from -0.64y+\frac{16\sqrt{3}}{75}x=-\frac{176\sqrt{3}}{15} by subtracting like terms on each side of the equal sign.

$$-0.64y+0.64y+\frac{16\sqrt{3}}{75}x-x=-\frac{176\sqrt{3}}{15}-13.44$$

Add -\frac{16y}{25} to \frac{16y}{25}. Terms -\frac{16y}{25} and \frac{16y}{25} cancel out, leaving an equation with only one variable that can be solved.

$$\frac{16\sqrt{3}}{75}x-x=-\frac{176\sqrt{3}}{15}-13.44$$

Add \frac{16\sqrt{3}x}{75} to -x.

$$\left(\frac{16\sqrt{3}}{75}-1\right)x=-\frac{176\sqrt{3}}{15}-13.44$$

Add -\frac{176\sqrt{3}}{15} to -13.44.

$$\left(\frac{16\sqrt{3}}{75}-1\right)x=-\frac{176\sqrt{3}}{15}-\frac{336}{25}$$

Divide both sides by \frac{16\sqrt{3}}{75}-1.

$$x=\frac{27376\sqrt{3}+39280}{1619}$$

Substitute \frac{39280+27376\sqrt{3}}{1619} for x in -0.64y+x=13.44. Because the resulting equation contains only one variable, you can solve for y directly.

$$-0.64y+\frac{27376\sqrt{3}+39280}{1619}=13.44$$

Subtract \frac{27376\sqrt{3}+39280}{1619} from both sides of the equation.

$$-0.64y=-\frac{27376\sqrt{3}}{1619}-\frac{438016}{40475}$$

Divide both sides of the equation by -0.64, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=\frac{42775\sqrt{3}+27376}{1619}$$

The system is now solved.

$$y=\frac{42775\sqrt{3}+27376}{1619},x=\frac{27376\sqrt{3}+39280}{1619}$$