Solve {l}{2x+7y-18=4(x+y)}{5x-4y-13=2(x-y)}

$$\left. \begin{array} { l } { 2 x + 7 y - 18 = 4 ( x + y ) } \\ { 5 x - 4 y - 13 = 2 ( x - y ) } \end{array} \right.$$

Solve for x, y

$$x=15$$

$$y=16$$

Steps Using Substitution

Consider the first equation. Use the distributive property to multiply 4 by x+y.

$$2x+7y-18=4x+4y$$

Subtract 4x from both sides.

$$2x+7y-18-4x=4y$$

Combine 2x and -4x to get -2x.

$$-2x+7y-18=4y$$

Subtract 4y from both sides.

$$-2x+7y-18-4y=0$$

Combine 7y and -4y to get 3y.

$$-2x+3y-18=0$$

Add 18 to both sides. Anything plus zero gives itself.

$$-2x+3y=18$$

Consider the second equation. Use the distributive property to multiply 2 by x-y.

$$5x-4y-13=2x-2y$$

Subtract 2x from both sides.

$$5x-4y-13-2x=-2y$$

Combine 5x and -2x to get 3x.

$$3x-4y-13=-2y$$

Add 2y to both sides.

$$3x-4y-13+2y=0$$

Combine -4y and 2y to get -2y.

$$3x-2y-13=0$$

Add 13 to both sides. Anything plus zero gives itself.

$$3x-2y=13$$

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.

$$-2x+3y=18,3x-2y=13$$

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

$$-2x+3y=18$$

Subtract 3y from both sides of the equation.

$$-2x=-3y+18$$

Divide both sides by -2.

$$x=-\frac{1}{2}\left(-3y+18\right)$$

Multiply -\frac{1}{2} times -3y+18.

$$x=\frac{3}{2}y-9$$

Substitute -9+\frac{3y}{2} for x in the other equation, 3x-2y=13.

$$3\left(\frac{3}{2}y-9\right)-2y=13$$

Multiply 3 times -9+\frac{3y}{2}.

$$\frac{9}{2}y-27-2y=13$$

Add \frac{9y}{2} to -2y.

$$\frac{5}{2}y-27=13$$

Add 27 to both sides of the equation.

$$\frac{5}{2}y=40$$

Divide both sides of the equation by \frac{5}{2}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=16$$

Substitute 16 for y in x=\frac{3}{2}y-9. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{3}{2}\times 16-9$$

Multiply \frac{3}{2} times 16.

$$x=24-9$$

Add -9 to 24.

$$x=15$$

The system is now solved.

$$x=15,y=16$$

Steps Using Matrices

Consider the first equation. Use the distributive property to multiply 4 by x+y.

$$2x+7y-18=4x+4y$$

Subtract 4x from both sides.

$$2x+7y-18-4x=4y$$

Combine 2x and -4x to get -2x.

$$-2x+7y-18=4y$$

Subtract 4y from both sides.

$$-2x+7y-18-4y=0$$

Combine 7y and -4y to get 3y.

$$-2x+3y-18=0$$

Add 18 to both sides. Anything plus zero gives itself.

$$-2x+3y=18$$

Consider the second equation. Use the distributive property to multiply 2 by x-y.

$$5x-4y-13=2x-2y$$

Subtract 2x from both sides.

$$5x-4y-13-2x=-2y$$

Combine 5x and -2x to get 3x.

$$3x-4y-13=-2y$$

Add 2y to both sides.

$$3x-4y-13+2y=0$$

Combine -4y and 2y to get -2y.

$$3x-2y-13=0$$

Add 13 to both sides. Anything plus zero gives itself.

$$3x-2y=13$$

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

$$-2x+3y=18,3x-2y=13$$

Write the equations in matrix form.

$$\left(\begin{matrix}-2&3\\3&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}18\\13\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}-2&3\\3&-2\end{matrix}\right).

$$inverse(\left(\begin{matrix}-2&3\\3&-2\end{matrix}\right))\left(\begin{matrix}-2&3\\3&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&3\\3&-2\end{matrix}\right))\left(\begin{matrix}18\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&3\\3&-2\end{matrix}\right))\left(\begin{matrix}18\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&3\\3&-2\end{matrix}\right))\left(\begin{matrix}18\\13\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{-2\left(-2\right)-3\times 3}&-\frac{3}{-2\left(-2\right)-3\times 3}\\-\frac{3}{-2\left(-2\right)-3\times 3}&-\frac{2}{-2\left(-2\right)-3\times 3}\end{matrix}\right)\left(\begin{matrix}18\\13\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{5}&\frac{3}{5}\\\frac{3}{5}&\frac{2}{5}\end{matrix}\right)\left(\begin{matrix}18\\13\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{5}\times 18+\frac{3}{5}\times 13\\\frac{3}{5}\times 18+\frac{2}{5}\times 13\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}15\\16\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=15,y=16$$

Steps Using Elimination

Consider the first equation. Use the distributive property to multiply 4 by x+y.

$$2x+7y-18=4x+4y$$

Subtract 4x from both sides.

$$2x+7y-18-4x=4y$$

Combine 2x and -4x to get -2x.

$$-2x+7y-18=4y$$

Subtract 4y from both sides.

$$-2x+7y-18-4y=0$$

Combine 7y and -4y to get 3y.

$$-2x+3y-18=0$$

Add 18 to both sides. Anything plus zero gives itself.

$$-2x+3y=18$$

Consider the second equation. Use the distributive property to multiply 2 by x-y.

$$5x-4y-13=2x-2y$$

Subtract 2x from both sides.

$$5x-4y-13-2x=-2y$$

Combine 5x and -2x to get 3x.

$$3x-4y-13=-2y$$

Add 2y to both sides.

$$3x-4y-13+2y=0$$

Combine -4y and 2y to get -2y.

$$3x-2y-13=0$$

Add 13 to both sides. Anything plus zero gives itself.

$$3x-2y=13$$

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.

$$-2x+3y=18,3x-2y=13$$

To make -2x and 3x equal, multiply all terms on each side of the first equation by 3 and all terms on each side of the second by -2.

$$3\left(-2\right)x+3\times 3y=3\times 18,-2\times 3x-2\left(-2\right)y=-2\times 13$$

Simplify.

$$-6x+9y=54,-6x+4y=-26$$

Subtract -6x+4y=-26 from -6x+9y=54 by subtracting like terms on each side of the equal sign.

$$-6x+6x+9y-4y=54+26$$

Add -6x to 6x. Terms -6x and 6x cancel out, leaving an equation with only one variable that can be solved.

$$9y-4y=54+26$$

Add 9y to -4y.

$$5y=54+26$$

Add 54 to 26.

$$5y=80$$

Divide both sides by 5.

$$y=16$$

Substitute 16 for y in 3x-2y=13. Because the resulting equation contains only one variable, you can solve for x directly.

$$3x-2\times 16=13$$

Multiply -2 times 16.

$$3x-32=13$$

Add 32 to both sides of the equation.

$$3x=45$$

Divide both sides by 3.

$$x=15$$

The system is now solved.

$$x=15,y=16$$