Solve {l}{14x+3y=-12}{15x+4y=-15}

$$\left. \begin{array} { l } { 14 x + 3 y = - 12 } \\ { 15 x + 4 y = - 15 } \end{array} \right.$$

Solve for x, y

$$x=-\frac{3}{11}\approx -0.272727273$$

$$y = -\frac{30}{11} = -2\frac{8}{11} \approx -2.727272727$$

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.

$$14x+3y=-12,15x+4y=-15$$

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

$$14x+3y=-12$$

Subtract 3y from both sides of the equation.

$$14x=-3y-12$$

Divide both sides by 14.

$$x=\frac{1}{14}\left(-3y-12\right)$$

Multiply \frac{1}{14} times -3y-12.

$$x=-\frac{3}{14}y-\frac{6}{7}$$

Substitute -\frac{3y}{14}-\frac{6}{7} for x in the other equation, 15x+4y=-15.

$$15\left(-\frac{3}{14}y-\frac{6}{7}\right)+4y=-15$$

Multiply 15 times -\frac{3y}{14}-\frac{6}{7}.

$$-\frac{45}{14}y-\frac{90}{7}+4y=-15$$

Add -\frac{45y}{14} to 4y.

$$\frac{11}{14}y-\frac{90}{7}=-15$$

Add \frac{90}{7} to both sides of the equation.

$$\frac{11}{14}y=-\frac{15}{7}$$

Divide both sides of the equation by \frac{11}{14}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=-\frac{30}{11}$$

Substitute -\frac{30}{11} for y in x=-\frac{3}{14}y-\frac{6}{7}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{3}{14}\left(-\frac{30}{11}\right)-\frac{6}{7}$$

Multiply -\frac{3}{14} times -\frac{30}{11} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{45}{77}-\frac{6}{7}$$

Add -\frac{6}{7} to \frac{45}{77} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{3}{11}$$

The system is now solved.

$$x=-\frac{3}{11},y=-\frac{30}{11}$$

Steps Using Matrices

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

$$14x+3y=-12,15x+4y=-15$$

Write the equations in matrix form.

$$\left(\begin{matrix}14&3\\15&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-12\\-15\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}14&3\\15&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}14&3\\15&4\end{matrix}\right))\left(\begin{matrix}14&3\\15&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}14&3\\15&4\end{matrix}\right))\left(\begin{matrix}-12\\-15\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}14&3\\15&4\end{matrix}\right))\left(\begin{matrix}-12\\-15\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}14&3\\15&4\end{matrix}\right))\left(\begin{matrix}-12\\-15\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{14\times 4-3\times 15}&-\frac{3}{14\times 4-3\times 15}\\-\frac{15}{14\times 4-3\times 15}&\frac{14}{14\times 4-3\times 15}\end{matrix}\right)\left(\begin{matrix}-12\\-15\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{11}&-\frac{3}{11}\\-\frac{15}{11}&\frac{14}{11}\end{matrix}\right)\left(\begin{matrix}-12\\-15\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{11}\left(-12\right)-\frac{3}{11}\left(-15\right)\\-\frac{15}{11}\left(-12\right)+\frac{14}{11}\left(-15\right)\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{11}\\-\frac{30}{11}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=-\frac{3}{11},y=-\frac{30}{11}$$

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.

$$14x+3y=-12,15x+4y=-15$$

To make 14x 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 14.

$$15\times 14x+15\times 3y=15\left(-12\right),14\times 15x+14\times 4y=14\left(-15\right)$$

Simplify.

$$210x+45y=-180,210x+56y=-210$$

Subtract 210x+56y=-210 from 210x+45y=-180 by subtracting like terms on each side of the equal sign.

$$210x-210x+45y-56y=-180+210$$

Add 210x to -210x. Terms 210x and -210x cancel out, leaving an equation with only one variable that can be solved.

$$45y-56y=-180+210$$

Add 45y to -56y.

$$-11y=-180+210$$

Add -180 to 210.

$$-11y=30$$

Divide both sides by -11.

$$y=-\frac{30}{11}$$

Substitute -\frac{30}{11} for y in 15x+4y=-15. Because the resulting equation contains only one variable, you can solve for x directly.

$$15x+4\left(-\frac{30}{11}\right)=-15$$

Multiply 4 times -\frac{30}{11}.

$$15x-\frac{120}{11}=-15$$

Add \frac{120}{11} to both sides of the equation.

$$15x=-\frac{45}{11}$$

Divide both sides by 15.

$$x=-\frac{3}{11}$$

The system is now solved.

$$x=-\frac{3}{11},y=-\frac{30}{11}$$