Solve {l}{20x+15y=540}{4x+5y=14}

$$\left\{ \begin{array} { l } { 20 x + 15 y = 540 } \\ { 4 x + 5 y = 14 } \end{array} \right.$$

Solve for x, y

$$x = \frac{249}{4} = 62\frac{1}{4} = 62.25$$

$$y=-47$$

Steps Using Substitution

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.

$$20x+15y=540,4x+5y=14$$

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

$$20x+15y=540$$

Subtract 15y from both sides of the equation.

$$20x=-15y+540$$

Divide both sides by 20.

$$x=\frac{1}{20}\left(-15y+540\right)$$

Multiply \frac{1}{20} times -15y+540.

$$x=-\frac{3}{4}y+27$$

Substitute -\frac{3y}{4}+27 for x in the other equation, 4x+5y=14.

$$4\left(-\frac{3}{4}y+27\right)+5y=14$$

Multiply 4 times -\frac{3y}{4}+27.

$$-3y+108+5y=14$$

Add -3y to 5y.

$$2y+108=14$$

Subtract 108 from both sides of the equation.

$$2y=-94$$

Divide both sides by 2.

$$y=-47$$

Substitute -47 for y in x=-\frac{3}{4}y+27. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{3}{4}\left(-47\right)+27$$

Multiply -\frac{3}{4} times -47.

$$x=\frac{141}{4}+27$$

Add 27 to \frac{141}{4}.

$$x=\frac{249}{4}$$

The system is now solved.

$$x=\frac{249}{4},y=-47$$

Steps Using Matrices

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

$$20x+15y=540,4x+5y=14$$

Write the equations in matrix form.

$$\left(\begin{matrix}20&15\\4&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}540\\14\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}20&15\\4&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}20&15\\4&5\end{matrix}\right))\left(\begin{matrix}20&15\\4&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}20&15\\4&5\end{matrix}\right))\left(\begin{matrix}540\\14\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}20&15\\4&5\end{matrix}\right))\left(\begin{matrix}540\\14\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}20&15\\4&5\end{matrix}\right))\left(\begin{matrix}540\\14\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{20\times 5-15\times 4}&-\frac{15}{20\times 5-15\times 4}\\-\frac{4}{20\times 5-15\times 4}&\frac{20}{20\times 5-15\times 4}\end{matrix}\right)\left(\begin{matrix}540\\14\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{8}&-\frac{3}{8}\\-\frac{1}{10}&\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}540\\14\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{8}\times 540-\frac{3}{8}\times 14\\-\frac{1}{10}\times 540+\frac{1}{2}\times 14\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{249}{4}\\-47\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{249}{4},y=-47$$

Steps Using Elimination

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.

$$20x+15y=540,4x+5y=14$$

To make 20x and 4x equal, multiply all terms on each side of the first equation by 4 and all terms on each side of the second by 20.

$$4\times 20x+4\times 15y=4\times 540,20\times 4x+20\times 5y=20\times 14$$

Simplify.

$$80x+60y=2160,80x+100y=280$$

Subtract 80x+100y=280 from 80x+60y=2160 by subtracting like terms on each side of the equal sign.

$$80x-80x+60y-100y=2160-280$$

Add 80x to -80x. Terms 80x and -80x cancel out, leaving an equation with only one variable that can be solved.

$$60y-100y=2160-280$$

Add 60y to -100y.

$$-40y=2160-280$$

Add 2160 to -280.

$$-40y=1880$$

Divide both sides by -40.

$$y=-47$$

Substitute -47 for y in 4x+5y=14. Because the resulting equation contains only one variable, you can solve for x directly.

$$4x+5\left(-47\right)=14$$

Multiply 5 times -47.

$$4x-235=14$$

Add 235 to both sides of the equation.

$$4x=249$$

Divide both sides by 4.

$$x=\frac{249}{4}$$

The system is now solved.

$$x=\frac{249}{4},y=-47$$