Solve {l}{22x+13y=140}{19x+17y=144}

$$\left. \begin{array} { l } { 22 x + 13 y = 140 } \\ { 19 x + 17 y = 144 } \end{array} \right.$$

Solve for x, y

$$x=4$$

$$y=4$$

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.

$$22x+13y=140,19x+17y=144$$

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

$$22x+13y=140$$

Subtract 13y from both sides of the equation.

$$22x=-13y+140$$

Divide both sides by 22.

$$x=\frac{1}{22}\left(-13y+140\right)$$

Multiply \frac{1}{22} times -13y+140.

$$x=-\frac{13}{22}y+\frac{70}{11}$$

Substitute -\frac{13y}{22}+\frac{70}{11} for x in the other equation, 19x+17y=144.

$$19\left(-\frac{13}{22}y+\frac{70}{11}\right)+17y=144$$

Multiply 19 times -\frac{13y}{22}+\frac{70}{11}.

$$-\frac{247}{22}y+\frac{1330}{11}+17y=144$$

Add -\frac{247y}{22} to 17y.

$$\frac{127}{22}y+\frac{1330}{11}=144$$

Subtract \frac{1330}{11} from both sides of the equation.

$$\frac{127}{22}y=\frac{254}{11}$$

Divide both sides of the equation by \frac{127}{22}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=4$$

Substitute 4 for y in x=-\frac{13}{22}y+\frac{70}{11}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{13}{22}\times 4+\frac{70}{11}$$

Multiply -\frac{13}{22} times 4.

$$x=\frac{-26+70}{11}$$

Add \frac{70}{11} to -\frac{26}{11} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=4$$

The system is now solved.

$$x=4,y=4$$

Steps Using Matrices

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

$$22x+13y=140,19x+17y=144$$

Write the equations in matrix form.

$$\left(\begin{matrix}22&13\\19&17\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}140\\144\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}22&13\\19&17\end{matrix}\right).

$$inverse(\left(\begin{matrix}22&13\\19&17\end{matrix}\right))\left(\begin{matrix}22&13\\19&17\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}22&13\\19&17\end{matrix}\right))\left(\begin{matrix}140\\144\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}22&13\\19&17\end{matrix}\right))\left(\begin{matrix}140\\144\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}22&13\\19&17\end{matrix}\right))\left(\begin{matrix}140\\144\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{17}{22\times 17-13\times 19}&-\frac{13}{22\times 17-13\times 19}\\-\frac{19}{22\times 17-13\times 19}&\frac{22}{22\times 17-13\times 19}\end{matrix}\right)\left(\begin{matrix}140\\144\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{17}{127}&-\frac{13}{127}\\-\frac{19}{127}&\frac{22}{127}\end{matrix}\right)\left(\begin{matrix}140\\144\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{17}{127}\times 140-\frac{13}{127}\times 144\\-\frac{19}{127}\times 140+\frac{22}{127}\times 144\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\4\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=4,y=4$$

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.

$$22x+13y=140,19x+17y=144$$

To make 22x and 19x equal, multiply all terms on each side of the first equation by 19 and all terms on each side of the second by 22.

$$19\times 22x+19\times 13y=19\times 140,22\times 19x+22\times 17y=22\times 144$$

Simplify.

$$418x+247y=2660,418x+374y=3168$$

Subtract 418x+374y=3168 from 418x+247y=2660 by subtracting like terms on each side of the equal sign.

$$418x-418x+247y-374y=2660-3168$$

Add 418x to -418x. Terms 418x and -418x cancel out, leaving an equation with only one variable that can be solved.

$$247y-374y=2660-3168$$

Add 247y to -374y.

$$-127y=2660-3168$$

Add 2660 to -3168.

$$-127y=-508$$

Divide both sides by -127.

$$y=4$$

Substitute 4 for y in 19x+17y=144. Because the resulting equation contains only one variable, you can solve for x directly.

$$19x+17\times 4=144$$

Multiply 17 times 4.

$$19x+68=144$$

Subtract 68 from both sides of the equation.

$$19x=76$$

Divide both sides by 19.

$$x=4$$

The system is now solved.

$$x=4,y=4$$