Solve {l}{3/5=x/y}{x-2/y=4/11}

$$\left. \begin{array} { l } { \frac { 3 } { 5 } = \frac { x } { y } } \\ { \frac { x - 2 } { y } = \frac { 4 } { 11 } } \end{array} \right.$$

Solve for x, y

$$x = \frac{66}{13} = 5\frac{1}{13} \approx 5.076923077$$

$$y = \frac{110}{13} = 8\frac{6}{13} \approx 8.461538462$$

Steps Using Substitution

Consider the first equation. Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 5y, the least common multiple of 5,y.

$$3y=5x$$

Divide both sides by 3.

$$y=\frac{1}{3}\times 5x$$

Multiply \frac{1}{3} times 5x.

$$y=\frac{5}{3}x$$

Substitute \frac{5x}{3} for y in the other equation, -4y+11x=22.

$$-4\times \frac{5}{3}x+11x=22$$

Multiply -4 times \frac{5x}{3}.

$$-\frac{20}{3}x+11x=22$$

Add -\frac{20x}{3} to 11x.

$$\frac{13}{3}x=22$$

Divide both sides of the equation by \frac{13}{3}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$x=\frac{66}{13}$$

Substitute \frac{66}{13} for x in y=\frac{5}{3}x. Because the resulting equation contains only one variable, you can solve for y directly.

$$y=\frac{5}{3}\times \frac{66}{13}$$

Multiply \frac{5}{3} times \frac{66}{13} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$y=\frac{110}{13}$$

The system is now solved.

$$y=\frac{110}{13},x=\frac{66}{13}$$

Steps Using Matrices

Consider the first equation. Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 5y, the least common multiple of 5,y.

$$3y=5x$$

Subtract 5x from both sides.

$$3y-5x=0$$

Consider the second equation. Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 11y, the least common multiple of y,11.

$$11\left(x-2\right)=4y$$

Use the distributive property to multiply 11 by x-2.

$$11x-22=4y$$

Subtract 4y from both sides.

$$11x-22-4y=0$$

Add 22 to both sides. Anything plus zero gives itself.

$$11x-4y=22$$

Put the equations in standard form and then use matrices to solve the system of equations.

$$3y-5x=0,-4y+11x=22$$

Write the equations in matrix form.

$$\left(\begin{matrix}3&-5\\-4&11\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}0\\22\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}3&-5\\-4&11\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-5\\-4&11\end{matrix}\right))\left(\begin{matrix}3&-5\\-4&11\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}3&-5\\-4&11\end{matrix}\right))\left(\begin{matrix}0\\22\end{matrix}\right)$$

The product of a matrix and its inverse is the identity matrix.

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}3&-5\\-4&11\end{matrix}\right))\left(\begin{matrix}0\\22\end{matrix}\right)$$

Multiply the matrices on the left hand side of the equal sign.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}3&-5\\-4&11\end{matrix}\right))\left(\begin{matrix}0\\22\end{matrix}\right)$$

For the 2\times 2 matrix \left(\begin{matrix}a&b\\c&d\end{matrix}\right), the inverse matrix is \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), so the matrix equation can be rewritten as a matrix multiplication problem.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{11}{3\times 11-\left(-5\left(-4\right)\right)}&-\frac{-5}{3\times 11-\left(-5\left(-4\right)\right)}\\-\frac{-4}{3\times 11-\left(-5\left(-4\right)\right)}&\frac{3}{3\times 11-\left(-5\left(-4\right)\right)}\end{matrix}\right)\left(\begin{matrix}0\\22\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{11}{13}&\frac{5}{13}\\\frac{4}{13}&\frac{3}{13}\end{matrix}\right)\left(\begin{matrix}0\\22\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{5}{13}\times 22\\\frac{3}{13}\times 22\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{110}{13}\\\frac{66}{13}\end{matrix}\right)$$

Extract the matrix elements y and x.

$$y=\frac{110}{13},x=\frac{66}{13}$$

Steps Using Elimination

Consider the first equation. Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 5y, the least common multiple of 5,y.

$$3y=5x$$

Subtract 5x from both sides.

$$3y-5x=0$$

Consider the second equation. Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 11y, the least common multiple of y,11.

$$11\left(x-2\right)=4y$$

Use the distributive property to multiply 11 by x-2.

$$11x-22=4y$$

Subtract 4y from both sides.

$$11x-22-4y=0$$

Add 22 to both sides. Anything plus zero gives itself.

$$11x-4y=22$$

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.

$$3y-5x=0,-4y+11x=22$$

To make 3y and -4y equal, multiply all terms on each side of the first equation by -4 and all terms on each side of the second by 3.

$$-4\times 3y-4\left(-5\right)x=0,3\left(-4\right)y+3\times 11x=3\times 22$$

Simplify.

$$-12y+20x=0,-12y+33x=66$$

Subtract -12y+33x=66 from -12y+20x=0 by subtracting like terms on each side of the equal sign.

$$-12y+12y+20x-33x=-66$$

Add -12y to 12y. Terms -12y and 12y cancel out, leaving an equation with only one variable that can be solved.

$$20x-33x=-66$$

Add 20x to -33x.

$$-13x=-66$$

Divide both sides by -13.

$$x=\frac{66}{13}$$

Substitute \frac{66}{13} for x in -4y+11x=22. Because the resulting equation contains only one variable, you can solve for y directly.

$$-4y+11\times \frac{66}{13}=22$$

Multiply 11 times \frac{66}{13}.

$$-4y+\frac{726}{13}=22$$

Subtract \frac{726}{13} from both sides of the equation.

$$-4y=-\frac{440}{13}$$

Divide both sides by -4.

$$y=\frac{110}{13}$$

The system is now solved.

$$y=\frac{110}{13},x=\frac{66}{13}$$