Solve {l}{x-1=y}{1200/x=5/4*800/y}

$$\left\{ \begin{array} { l } { x - 1 = y } \\ { \frac { 1200 } { x } = \frac { 5 } { 4 } \cdot \frac { 800 } { y } } \end{array} \right.$$

Solve for x, y

$$x=6$$

$$y=5$$

Steps Using Substitution

Consider the first equation. Subtract y from both sides.

$$x-1-y=0$$

Add 1 to both sides. Anything plus zero gives itself.

$$x-y=1$$

Consider the second equation. Variable x cannot be equal to 0 since division by zero is not defined. Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 4xy, the least common multiple of x,4,y.

$$4y\times 1200=\frac{5}{4}\times 4x\times 800$$

Multiply 4 and 1200 to get 4800.

$$4800y=\frac{5}{4}\times 4x\times 800$$

Multiply \frac{5}{4} and 4 to get 5.

$$4800y=5x\times 800$$

Multiply 5 and 800 to get 4000.

$$4800y=4000x$$

Subtract 4000x from both sides.

$$4800y-4000x=0$$

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.

$$x-y=1,-4000x+4800y=0$$

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

$$x-y=1$$

Add y to both sides of the equation.

$$x=y+1$$

Substitute 1+y for x in the other equation, -4000x+4800y=0.

$$-4000\left(y+1\right)+4800y=0$$

Multiply -4000 times y+1.

$$-4000y-4000+4800y=0$$

Add -4000y to 4800y.

$$800y-4000=0$$

Add 4000 to both sides of the equation.

$$800y=4000$$

Divide both sides by 800.

$$y=5$$

Substitute 5 for y in x=y+1. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=5+1$$

Add 1 to 5.

$$x=6$$

The system is now solved.

$$x=6,y=5$$

Steps Using Matrices

Consider the first equation. Subtract y from both sides.

$$x-1-y=0$$

Add 1 to both sides. Anything plus zero gives itself.

$$x-y=1$$

Consider the second equation. Variable x cannot be equal to 0 since division by zero is not defined. Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 4xy, the least common multiple of x,4,y.

$$4y\times 1200=\frac{5}{4}\times 4x\times 800$$

Multiply 4 and 1200 to get 4800.

$$4800y=\frac{5}{4}\times 4x\times 800$$

Multiply \frac{5}{4} and 4 to get 5.

$$4800y=5x\times 800$$

Multiply 5 and 800 to get 4000.

$$4800y=4000x$$

Subtract 4000x from both sides.

$$4800y-4000x=0$$

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

$$x-y=1,-4000x+4800y=0$$

Write the equations in matrix form.

$$\left(\begin{matrix}1&-1\\-4000&4800\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\0\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&-1\\-4000&4800\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-1\\-4000&4800\end{matrix}\right))\left(\begin{matrix}1&-1\\-4000&4800\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\-4000&4800\end{matrix}\right))\left(\begin{matrix}1\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\-4000&4800\end{matrix}\right))\left(\begin{matrix}1\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\-4000&4800\end{matrix}\right))\left(\begin{matrix}1\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4800}{4800-\left(-\left(-4000\right)\right)}&-\frac{-1}{4800-\left(-\left(-4000\right)\right)}\\-\frac{-4000}{4800-\left(-\left(-4000\right)\right)}&\frac{1}{4800-\left(-\left(-4000\right)\right)}\end{matrix}\right)\left(\begin{matrix}1\\0\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6&\frac{1}{800}\\5&\frac{1}{800}\end{matrix}\right)\left(\begin{matrix}1\\0\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\5\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=6,y=5$$

Steps Using Elimination

Consider the first equation. Subtract y from both sides.

$$x-1-y=0$$

Add 1 to both sides. Anything plus zero gives itself.

$$x-y=1$$

Consider the second equation. Variable x cannot be equal to 0 since division by zero is not defined. Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 4xy, the least common multiple of x,4,y.

$$4y\times 1200=\frac{5}{4}\times 4x\times 800$$

Multiply 4 and 1200 to get 4800.

$$4800y=\frac{5}{4}\times 4x\times 800$$

Multiply \frac{5}{4} and 4 to get 5.

$$4800y=5x\times 800$$

Multiply 5 and 800 to get 4000.

$$4800y=4000x$$

Subtract 4000x from both sides.

$$4800y-4000x=0$$

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.

$$x-y=1,-4000x+4800y=0$$

To make x and -4000x equal, multiply all terms on each side of the first equation by -4000 and all terms on each side of the second by 1.

$$-4000x-4000\left(-1\right)y=-4000,-4000x+4800y=0$$

Simplify.

$$-4000x+4000y=-4000,-4000x+4800y=0$$

Subtract -4000x+4800y=0 from -4000x+4000y=-4000 by subtracting like terms on each side of the equal sign.

$$-4000x+4000x+4000y-4800y=-4000$$

Add -4000x to 4000x. Terms -4000x and 4000x cancel out, leaving an equation with only one variable that can be solved.

$$4000y-4800y=-4000$$

Add 4000y to -4800y.

$$-800y=-4000$$

Divide both sides by -800.

$$y=5$$

Substitute 5 for y in -4000x+4800y=0. Because the resulting equation contains only one variable, you can solve for x directly.

$$-4000x+4800\times 5=0$$

Multiply 4800 times 5.

$$-4000x+24000=0$$

Subtract 24000 from both sides of the equation.

$$-4000x=-24000$$

Divide both sides by -4000.

$$x=6$$

The system is now solved.

$$x=6,y=5$$