Solve {l}{6x+5y=100}{15x+20y=310}

$$\left. \begin{array} { l } { 6 x + 5 y = 100 } \\ { 15 x + 20 y = 310 } \end{array} \right.$$

Solve for x, y

$$x=10$$

$$y=8$$

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.

$$6x+5y=100,15x+20y=310$$

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

$$6x+5y=100$$

Subtract 5y from both sides of the equation.

$$6x=-5y+100$$

Divide both sides by 6.

$$x=\frac{1}{6}\left(-5y+100\right)$$

Multiply \frac{1}{6} times -5y+100.

$$x=-\frac{5}{6}y+\frac{50}{3}$$

Substitute -\frac{5y}{6}+\frac{50}{3} for x in the other equation, 15x+20y=310.

$$15\left(-\frac{5}{6}y+\frac{50}{3}\right)+20y=310$$

Multiply 15 times -\frac{5y}{6}+\frac{50}{3}.

$$-\frac{25}{2}y+250+20y=310$$

Add -\frac{25y}{2} to 20y.

$$\frac{15}{2}y+250=310$$

Subtract 250 from both sides of the equation.

$$\frac{15}{2}y=60$$

Divide both sides of the equation by \frac{15}{2}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=8$$

Substitute 8 for y in x=-\frac{5}{6}y+\frac{50}{3}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{5}{6}\times 8+\frac{50}{3}$$

Multiply -\frac{5}{6} times 8.

$$x=\frac{-20+50}{3}$$

Add \frac{50}{3} to -\frac{20}{3} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=10$$

The system is now solved.

$$x=10,y=8$$

Steps Using Matrices

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

$$6x+5y=100,15x+20y=310$$

Write the equations in matrix form.

$$\left(\begin{matrix}6&5\\15&20\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}100\\310\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}6&5\\15&20\end{matrix}\right).

$$inverse(\left(\begin{matrix}6&5\\15&20\end{matrix}\right))\left(\begin{matrix}6&5\\15&20\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&5\\15&20\end{matrix}\right))\left(\begin{matrix}100\\310\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&5\\15&20\end{matrix}\right))\left(\begin{matrix}100\\310\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&5\\15&20\end{matrix}\right))\left(\begin{matrix}100\\310\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{20}{6\times 20-5\times 15}&-\frac{5}{6\times 20-5\times 15}\\-\frac{15}{6\times 20-5\times 15}&\frac{6}{6\times 20-5\times 15}\end{matrix}\right)\left(\begin{matrix}100\\310\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{9}&-\frac{1}{9}\\-\frac{1}{3}&\frac{2}{15}\end{matrix}\right)\left(\begin{matrix}100\\310\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{9}\times 100-\frac{1}{9}\times 310\\-\frac{1}{3}\times 100+\frac{2}{15}\times 310\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\8\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=10,y=8$$

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.

$$6x+5y=100,15x+20y=310$$

To make 6x and 15x equal, multiply all terms on each side of the first equation by 15 and all terms on each side of the second by 6.

$$15\times 6x+15\times 5y=15\times 100,6\times 15x+6\times 20y=6\times 310$$

Simplify.

$$90x+75y=1500,90x+120y=1860$$

Subtract 90x+120y=1860 from 90x+75y=1500 by subtracting like terms on each side of the equal sign.

$$90x-90x+75y-120y=1500-1860$$

Add 90x to -90x. Terms 90x and -90x cancel out, leaving an equation with only one variable that can be solved.

$$75y-120y=1500-1860$$

Add 75y to -120y.

$$-45y=1500-1860$$

Add 1500 to -1860.

$$-45y=-360$$

Divide both sides by -45.

$$y=8$$

Substitute 8 for y in 15x+20y=310. Because the resulting equation contains only one variable, you can solve for x directly.

$$15x+20\times 8=310$$

Multiply 20 times 8.

$$15x+160=310$$

Subtract 160 from both sides of the equation.

$$15x=150$$

Divide both sides by 15.

$$x=10$$

The system is now solved.

$$x=10,y=8$$