Solve {l}{5(2x+y)=2(x+2y)+19}{4(x-2x)=3(3x-4y)-10}

$$\left\{ \begin{array} { l } { 5 ( 2 x + y ) = 2 ( x + 2 y ) + 19 } \\ { 4 ( x - 2 x ) = 3 ( 3 x - 4 y ) - 10 } \end{array} \right.$$

Solve for x, y

$$x = \frac{238}{109} = 2\frac{20}{109} \approx 2.183486239$$

$$y = \frac{167}{109} = 1\frac{58}{109} \approx 1.532110092$$

Steps Using Substitution

Consider the first equation. Use the distributive property to multiply 5 by 2x+y.

$$10x+5y=2\left(x+2y\right)+19$$

Use the distributive property to multiply 2 by x+2y.

$$10x+5y=2x+4y+19$$

Subtract 2x from both sides.

$$10x+5y-2x=4y+19$$

Combine 10x and -2x to get 8x.

$$8x+5y=4y+19$$

Subtract 4y from both sides.

$$8x+5y-4y=19$$

Combine 5y and -4y to get y.

$$8x+y=19$$

Consider the second equation. Combine x and -2x to get -x.

$$4\left(-1\right)x=3\left(3x-4y\right)-10$$

Multiply 4 and -1 to get -4.

$$-4x=3\left(3x-4y\right)-10$$

Use the distributive property to multiply 3 by 3x-4y.

$$-4x=9x-12y-10$$

Subtract 9x from both sides.

$$-4x-9x=-12y-10$$

Combine -4x and -9x to get -13x.

$$-13x=-12y-10$$

Add 12y to both sides.

$$-13x+12y=-10$$

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.

$$8x+y=19,-13x+12y=-10$$

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

$$8x+y=19$$

Subtract y from both sides of the equation.

$$8x=-y+19$$

Divide both sides by 8.

$$x=\frac{1}{8}\left(-y+19\right)$$

Multiply \frac{1}{8} times -y+19.

$$x=-\frac{1}{8}y+\frac{19}{8}$$

Substitute \frac{-y+19}{8} for x in the other equation, -13x+12y=-10.

$$-13\left(-\frac{1}{8}y+\frac{19}{8}\right)+12y=-10$$

Multiply -13 times \frac{-y+19}{8}.

$$\frac{13}{8}y-\frac{247}{8}+12y=-10$$

Add \frac{13y}{8} to 12y.

$$\frac{109}{8}y-\frac{247}{8}=-10$$

Add \frac{247}{8} to both sides of the equation.

$$\frac{109}{8}y=\frac{167}{8}$$

Divide both sides of the equation by \frac{109}{8}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=\frac{167}{109}$$

Substitute \frac{167}{109} for y in x=-\frac{1}{8}y+\frac{19}{8}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{1}{8}\times \frac{167}{109}+\frac{19}{8}$$

Multiply -\frac{1}{8} times \frac{167}{109} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{167}{872}+\frac{19}{8}$$

Add \frac{19}{8} to -\frac{167}{872} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{238}{109}$$

The system is now solved.

$$x=\frac{238}{109},y=\frac{167}{109}$$

Steps Using Matrices

Consider the first equation. Use the distributive property to multiply 5 by 2x+y.

$$10x+5y=2\left(x+2y\right)+19$$

Use the distributive property to multiply 2 by x+2y.

$$10x+5y=2x+4y+19$$

Subtract 2x from both sides.

$$10x+5y-2x=4y+19$$

Combine 10x and -2x to get 8x.

$$8x+5y=4y+19$$

Subtract 4y from both sides.

$$8x+5y-4y=19$$

Combine 5y and -4y to get y.

$$8x+y=19$$

Consider the second equation. Combine x and -2x to get -x.

$$4\left(-1\right)x=3\left(3x-4y\right)-10$$

Multiply 4 and -1 to get -4.

$$-4x=3\left(3x-4y\right)-10$$

Use the distributive property to multiply 3 by 3x-4y.

$$-4x=9x-12y-10$$

Subtract 9x from both sides.

$$-4x-9x=-12y-10$$

Combine -4x and -9x to get -13x.

$$-13x=-12y-10$$

Add 12y to both sides.

$$-13x+12y=-10$$

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

$$8x+y=19,-13x+12y=-10$$

Write the equations in matrix form.

$$\left(\begin{matrix}8&1\\-13&12\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}19\\-10\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}8&1\\-13&12\end{matrix}\right).

$$inverse(\left(\begin{matrix}8&1\\-13&12\end{matrix}\right))\left(\begin{matrix}8&1\\-13&12\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&1\\-13&12\end{matrix}\right))\left(\begin{matrix}19\\-10\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&1\\-13&12\end{matrix}\right))\left(\begin{matrix}19\\-10\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&1\\-13&12\end{matrix}\right))\left(\begin{matrix}19\\-10\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{12}{8\times 12-\left(-13\right)}&-\frac{1}{8\times 12-\left(-13\right)}\\-\frac{-13}{8\times 12-\left(-13\right)}&\frac{8}{8\times 12-\left(-13\right)}\end{matrix}\right)\left(\begin{matrix}19\\-10\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{12}{109}&-\frac{1}{109}\\\frac{13}{109}&\frac{8}{109}\end{matrix}\right)\left(\begin{matrix}19\\-10\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{12}{109}\times 19-\frac{1}{109}\left(-10\right)\\\frac{13}{109}\times 19+\frac{8}{109}\left(-10\right)\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{238}{109}\\\frac{167}{109}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{238}{109},y=\frac{167}{109}$$

Steps Using Elimination

Consider the first equation. Use the distributive property to multiply 5 by 2x+y.

$$10x+5y=2\left(x+2y\right)+19$$

Use the distributive property to multiply 2 by x+2y.

$$10x+5y=2x+4y+19$$

Subtract 2x from both sides.

$$10x+5y-2x=4y+19$$

Combine 10x and -2x to get 8x.

$$8x+5y=4y+19$$

Subtract 4y from both sides.

$$8x+5y-4y=19$$

Combine 5y and -4y to get y.

$$8x+y=19$$

Consider the second equation. Combine x and -2x to get -x.

$$4\left(-1\right)x=3\left(3x-4y\right)-10$$

Multiply 4 and -1 to get -4.

$$-4x=3\left(3x-4y\right)-10$$

Use the distributive property to multiply 3 by 3x-4y.

$$-4x=9x-12y-10$$

Subtract 9x from both sides.

$$-4x-9x=-12y-10$$

Combine -4x and -9x to get -13x.

$$-13x=-12y-10$$

Add 12y to both sides.

$$-13x+12y=-10$$

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.

$$8x+y=19,-13x+12y=-10$$

To make 8x and -13x equal, multiply all terms on each side of the first equation by -13 and all terms on each side of the second by 8.

$$-13\times 8x-13y=-13\times 19,8\left(-13\right)x+8\times 12y=8\left(-10\right)$$

Simplify.

$$-104x-13y=-247,-104x+96y=-80$$

Subtract -104x+96y=-80 from -104x-13y=-247 by subtracting like terms on each side of the equal sign.

$$-104x+104x-13y-96y=-247+80$$

Add -104x to 104x. Terms -104x and 104x cancel out, leaving an equation with only one variable that can be solved.

$$-13y-96y=-247+80$$

Add -13y to -96y.

$$-109y=-247+80$$

Add -247 to 80.

$$-109y=-167$$

Divide both sides by -109.

$$y=\frac{167}{109}$$

Substitute \frac{167}{109} for y in -13x+12y=-10. Because the resulting equation contains only one variable, you can solve for x directly.

$$-13x+12\times \frac{167}{109}=-10$$

Multiply 12 times \frac{167}{109}.

$$-13x+\frac{2004}{109}=-10$$

Subtract \frac{2004}{109} from both sides of the equation.

$$-13x=-\frac{3094}{109}$$

Divide both sides by -13.

$$x=\frac{238}{109}$$

The system is now solved.

$$x=\frac{238}{109},y=\frac{167}{109}$$