Solve {l}{19x+15y=-5}{12x+8y=-12}{text{yes}}

$$\left. \begin{array} { l } { 19 x + 15 y = - 5 } \\ { 12 x + 8 y = - 12 } \\ { \text { yes } } \end{array} \right.$$

Solve for x, y

$$x=-5$$

$$y=6$$

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.

$$19x+15y=-5,12x+8y=-12$$

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

$$19x+15y=-5$$

Subtract 15y from both sides of the equation.

$$19x=-15y-5$$

Divide both sides by 19.

$$x=\frac{1}{19}\left(-15y-5\right)$$

Multiply \frac{1}{19} times -15y-5.

$$x=-\frac{15}{19}y-\frac{5}{19}$$

Substitute \frac{-15y-5}{19} for x in the other equation, 12x+8y=-12.

$$12\left(-\frac{15}{19}y-\frac{5}{19}\right)+8y=-12$$

Multiply 12 times \frac{-15y-5}{19}.

$$-\frac{180}{19}y-\frac{60}{19}+8y=-12$$

Add -\frac{180y}{19} to 8y.

$$-\frac{28}{19}y-\frac{60}{19}=-12$$

Add \frac{60}{19} to both sides of the equation.

$$-\frac{28}{19}y=-\frac{168}{19}$$

Divide both sides of the equation by -\frac{28}{19}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=6$$

Substitute 6 for y in x=-\frac{15}{19}y-\frac{5}{19}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{15}{19}\times 6-\frac{5}{19}$$

Multiply -\frac{15}{19} times 6.

$$x=\frac{-90-5}{19}$$

Add -\frac{5}{19} to -\frac{90}{19} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-5$$

The system is now solved.

$$x=-5,y=6$$

Steps Using Matrices

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

$$19x+15y=-5,12x+8y=-12$$

Write the equations in matrix form.

$$\left(\begin{matrix}19&15\\12&8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-5\\-12\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}19&15\\12&8\end{matrix}\right).

$$inverse(\left(\begin{matrix}19&15\\12&8\end{matrix}\right))\left(\begin{matrix}19&15\\12&8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}19&15\\12&8\end{matrix}\right))\left(\begin{matrix}-5\\-12\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}19&15\\12&8\end{matrix}\right))\left(\begin{matrix}-5\\-12\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}19&15\\12&8\end{matrix}\right))\left(\begin{matrix}-5\\-12\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{19\times 8-15\times 12}&-\frac{15}{19\times 8-15\times 12}\\-\frac{12}{19\times 8-15\times 12}&\frac{19}{19\times 8-15\times 12}\end{matrix}\right)\left(\begin{matrix}-5\\-12\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{7}&\frac{15}{28}\\\frac{3}{7}&-\frac{19}{28}\end{matrix}\right)\left(\begin{matrix}-5\\-12\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{7}\left(-5\right)+\frac{15}{28}\left(-12\right)\\\frac{3}{7}\left(-5\right)-\frac{19}{28}\left(-12\right)\end{matrix}\right)$$

Do the arithmetic.

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

Extract the matrix elements x and y.

$$x=-5,y=6$$

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.

$$19x+15y=-5,12x+8y=-12$$

To make 19x and 12x equal, multiply all terms on each side of the first equation by 12 and all terms on each side of the second by 19.

$$12\times 19x+12\times 15y=12\left(-5\right),19\times 12x+19\times 8y=19\left(-12\right)$$

Simplify.

$$228x+180y=-60,228x+152y=-228$$

Subtract 228x+152y=-228 from 228x+180y=-60 by subtracting like terms on each side of the equal sign.

$$228x-228x+180y-152y=-60+228$$

Add 228x to -228x. Terms 228x and -228x cancel out, leaving an equation with only one variable that can be solved.

$$180y-152y=-60+228$$

Add 180y to -152y.

$$28y=-60+228$$

Add -60 to 228.

$$28y=168$$

Divide both sides by 28.

$$y=6$$

Substitute 6 for y in 12x+8y=-12. Because the resulting equation contains only one variable, you can solve for x directly.

$$12x+8\times 6=-12$$

Multiply 8 times 6.

$$12x+48=-12$$

Subtract 48 from both sides of the equation.

$$12x=-60$$

Divide both sides by 12.

$$x=-5$$

The system is now solved.

$$x=-5,y=6$$