Solve {l}{6x-3y=15}{8x-10y=-70}

$$\left. \begin{array} { l } { 6 x - 3 y = 15 } \\ { 8 x - 10 y = - 70 } \end{array} \right.$$

Solve for x, y

$$x=10$$

$$y=15$$

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.

$$6x-3y=15,8x-10y=-70$$

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

$$6x-3y=15$$

Add 3y to both sides of the equation.

$$6x=3y+15$$

Divide both sides by 6.

$$x=\frac{1}{6}\left(3y+15\right)$$

Multiply \frac{1}{6} times 15+3y.

$$x=\frac{1}{2}y+\frac{5}{2}$$

Substitute \frac{5+y}{2} for x in the other equation, 8x-10y=-70.

$$8\left(\frac{1}{2}y+\frac{5}{2}\right)-10y=-70$$

Multiply 8 times \frac{5+y}{2}.

$$4y+20-10y=-70$$

Add 4y to -10y.

$$-6y+20=-70$$

Subtract 20 from both sides of the equation.

$$-6y=-90$$

Divide both sides by -6.

$$y=15$$

Substitute 15 for y in x=\frac{1}{2}y+\frac{5}{2}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{1}{2}\times 15+\frac{5}{2}$$

Multiply \frac{1}{2} times 15.

$$x=\frac{15+5}{2}$$

Add \frac{5}{2} to \frac{15}{2} 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.

$$x=10,y=15$$

Steps Using Matrices

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

$$6x-3y=15,8x-10y=-70$$

Write the equations in matrix form.

$$\left(\begin{matrix}6&-3\\8&-10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}15\\-70\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}6&-3\\8&-10\end{matrix}\right).

$$inverse(\left(\begin{matrix}6&-3\\8&-10\end{matrix}\right))\left(\begin{matrix}6&-3\\8&-10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&-3\\8&-10\end{matrix}\right))\left(\begin{matrix}15\\-70\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&-3\\8&-10\end{matrix}\right))\left(\begin{matrix}15\\-70\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&-3\\8&-10\end{matrix}\right))\left(\begin{matrix}15\\-70\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{10}{6\left(-10\right)-\left(-3\times 8\right)}&-\frac{-3}{6\left(-10\right)-\left(-3\times 8\right)}\\-\frac{8}{6\left(-10\right)-\left(-3\times 8\right)}&\frac{6}{6\left(-10\right)-\left(-3\times 8\right)}\end{matrix}\right)\left(\begin{matrix}15\\-70\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{18}&-\frac{1}{12}\\\frac{2}{9}&-\frac{1}{6}\end{matrix}\right)\left(\begin{matrix}15\\-70\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{18}\times 15-\frac{1}{12}\left(-70\right)\\\frac{2}{9}\times 15-\frac{1}{6}\left(-70\right)\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\15\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=10,y=15$$

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.

$$6x-3y=15,8x-10y=-70$$

To make 6x and 8x equal, multiply all terms on each side of the first equation by 8 and all terms on each side of the second by 6.

$$8\times 6x+8\left(-3\right)y=8\times 15,6\times 8x+6\left(-10\right)y=6\left(-70\right)$$

Simplify.

$$48x-24y=120,48x-60y=-420$$

Subtract 48x-60y=-420 from 48x-24y=120 by subtracting like terms on each side of the equal sign.

$$48x-48x-24y+60y=120+420$$

Add 48x to -48x. Terms 48x and -48x cancel out, leaving an equation with only one variable that can be solved.

$$-24y+60y=120+420$$

Add -24y to 60y.

$$36y=120+420$$

Add 120 to 420.

$$36y=540$$

Divide both sides by 36.

$$y=15$$

Substitute 15 for y in 8x-10y=-70. Because the resulting equation contains only one variable, you can solve for x directly.

$$8x-10\times 15=-70$$

Multiply -10 times 15.

$$8x-150=-70$$

Add 150 to both sides of the equation.

$$8x=80$$

Divide both sides by 8.

$$x=10$$

The system is now solved.

$$x=10,y=15$$