Solve {l}{9+16x-14y=0}{15+21x-16y=0}

$$\left\{ \begin{array} { l } { 9 + 16 x - 14 y = 0 } \\ { 15 + 21 x - 16 y = 0 } \end{array} \right.$$

Solve for x, y

$$x = -\frac{33}{19} = -1\frac{14}{19} \approx -1.736842105$$

$$y = -\frac{51}{38} = -1\frac{13}{38} \approx -1.342105263$$

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.

$$16x-14y+9=0,21x-16y+15=0$$

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

$$16x-14y+9=0$$

Subtract 9 from both sides of the equation.

$$16x-14y=-9$$

Add 14y to both sides of the equation.

$$16x=14y-9$$

Divide both sides by 16.

$$x=\frac{1}{16}\left(14y-9\right)$$

Multiply \frac{1}{16} times 14y-9.

$$x=\frac{7}{8}y-\frac{9}{16}$$

Substitute \frac{7y}{8}-\frac{9}{16} for x in the other equation, 21x-16y+15=0.

$$21\left(\frac{7}{8}y-\frac{9}{16}\right)-16y+15=0$$

Multiply 21 times \frac{7y}{8}-\frac{9}{16}.

$$\frac{147}{8}y-\frac{189}{16}-16y+15=0$$

Add \frac{147y}{8} to -16y.

$$\frac{19}{8}y-\frac{189}{16}+15=0$$

Add -\frac{189}{16} to 15.

$$\frac{19}{8}y+\frac{51}{16}=0$$

Subtract \frac{51}{16} from both sides of the equation.

$$\frac{19}{8}y=-\frac{51}{16}$$

Divide both sides of the equation by \frac{19}{8}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=-\frac{51}{38}$$

Substitute -\frac{51}{38} for y in x=\frac{7}{8}y-\frac{9}{16}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{7}{8}\left(-\frac{51}{38}\right)-\frac{9}{16}$$

Multiply \frac{7}{8} times -\frac{51}{38} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{357}{304}-\frac{9}{16}$$

Add -\frac{9}{16} to -\frac{357}{304} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{33}{19}$$

The system is now solved.

$$x=-\frac{33}{19},y=-\frac{51}{38}$$

Steps Using Matrices

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

$$16x-14y+9=0,21x-16y+15=0$$

Write the equations in matrix form.

$$\left(\begin{matrix}16&-14\\21&-16\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-9\\-15\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}16&-14\\21&-16\end{matrix}\right).

$$inverse(\left(\begin{matrix}16&-14\\21&-16\end{matrix}\right))\left(\begin{matrix}16&-14\\21&-16\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}16&-14\\21&-16\end{matrix}\right))\left(\begin{matrix}-9\\-15\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}16&-14\\21&-16\end{matrix}\right))\left(\begin{matrix}-9\\-15\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}16&-14\\21&-16\end{matrix}\right))\left(\begin{matrix}-9\\-15\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{16}{16\left(-16\right)-\left(-14\times 21\right)}&-\frac{-14}{16\left(-16\right)-\left(-14\times 21\right)}\\-\frac{21}{16\left(-16\right)-\left(-14\times 21\right)}&\frac{16}{16\left(-16\right)-\left(-14\times 21\right)}\end{matrix}\right)\left(\begin{matrix}-9\\-15\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{8}{19}&\frac{7}{19}\\-\frac{21}{38}&\frac{8}{19}\end{matrix}\right)\left(\begin{matrix}-9\\-15\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{8}{19}\left(-9\right)+\frac{7}{19}\left(-15\right)\\-\frac{21}{38}\left(-9\right)+\frac{8}{19}\left(-15\right)\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{33}{19}\\-\frac{51}{38}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=-\frac{33}{19},y=-\frac{51}{38}$$

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.

$$16x-14y+9=0,21x-16y+15=0$$

To make 16x and 21x equal, multiply all terms on each side of the first equation by 21 and all terms on each side of the second by 16.

$$21\times 16x+21\left(-14\right)y+21\times 9=0,16\times 21x+16\left(-16\right)y+16\times 15=0$$

Simplify.

$$336x-294y+189=0,336x-256y+240=0$$

Subtract 336x-256y+240=0 from 336x-294y+189=0 by subtracting like terms on each side of the equal sign.

$$336x-336x-294y+256y+189-240=0$$

Add 336x to -336x. Terms 336x and -336x cancel out, leaving an equation with only one variable that can be solved.

$$-294y+256y+189-240=0$$

Add -294y to 256y.

$$-38y+189-240=0$$

Add 189 to -240.

$$-38y-51=0$$

Add 51 to both sides of the equation.

$$-38y=51$$

Divide both sides by -38.

$$y=-\frac{51}{38}$$

Substitute -\frac{51}{38} for y in 21x-16y+15=0. Because the resulting equation contains only one variable, you can solve for x directly.

$$21x-16\left(-\frac{51}{38}\right)+15=0$$

Multiply -16 times -\frac{51}{38}.

$$21x+\frac{408}{19}+15=0$$

Add \frac{408}{19} to 15.

$$21x+\frac{693}{19}=0$$

Subtract \frac{693}{19} from both sides of the equation.

$$21x=-\frac{693}{19}$$

Divide both sides by 21.

$$x=-\frac{33}{19}$$

The system is now solved.

$$x=-\frac{33}{19},y=-\frac{51}{38}$$