Solve {l}{9x+4y/3-5x-11/2=13-y}{13x-7y=-8}

$$\left. \begin{array} { l } { \frac { 9 x + 4 y } { 3 } - \frac { 5 x - 11 } { 2 } = 13 - y } \\ { 13 x - 7 y = - 8 } \end{array} \right.$$

Solve for x, y

$$x=1$$

$$y=3$$

Steps Using Substitution

Consider the first equation. Multiply both sides of the equation by 6, the least common multiple of 3,2.

$$2\left(9x+4y\right)-3\left(5x-11\right)=78-6y$$

Use the distributive property to multiply 2 by 9x+4y.

$$18x+8y-3\left(5x-11\right)=78-6y$$

Use the distributive property to multiply -3 by 5x-11.

$$18x+8y-15x+33=78-6y$$

Combine 18x and -15x to get 3x.

$$3x+8y+33=78-6y$$

Add 6y to both sides.

$$3x+8y+33+6y=78$$

Combine 8y and 6y to get 14y.

$$3x+14y+33=78$$

Subtract 33 from both sides.

$$3x+14y=78-33$$

Subtract 33 from 78 to get 45.

$$3x+14y=45$$

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.

$$3x+14y=45,13x-7y=-8$$

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

$$3x+14y=45$$

Subtract 14y from both sides of the equation.

$$3x=-14y+45$$

Divide both sides by 3.

$$x=\frac{1}{3}\left(-14y+45\right)$$

Multiply \frac{1}{3} times -14y+45.

$$x=-\frac{14}{3}y+15$$

Substitute -\frac{14y}{3}+15 for x in the other equation, 13x-7y=-8.

$$13\left(-\frac{14}{3}y+15\right)-7y=-8$$

Multiply 13 times -\frac{14y}{3}+15.

$$-\frac{182}{3}y+195-7y=-8$$

Add -\frac{182y}{3} to -7y.

$$-\frac{203}{3}y+195=-8$$

Subtract 195 from both sides of the equation.

$$-\frac{203}{3}y=-203$$

Divide both sides of the equation by -\frac{203}{3}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=3$$

Substitute 3 for y in x=-\frac{14}{3}y+15. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{14}{3}\times 3+15$$

Multiply -\frac{14}{3} times 3.

$$x=-14+15$$

Add 15 to -14.

$$x=1$$

The system is now solved.

$$x=1,y=3$$

Steps Using Matrices

Consider the first equation. Multiply both sides of the equation by 6, the least common multiple of 3,2.

$$2\left(9x+4y\right)-3\left(5x-11\right)=78-6y$$

Use the distributive property to multiply 2 by 9x+4y.

$$18x+8y-3\left(5x-11\right)=78-6y$$

Use the distributive property to multiply -3 by 5x-11.

$$18x+8y-15x+33=78-6y$$

Combine 18x and -15x to get 3x.

$$3x+8y+33=78-6y$$

Add 6y to both sides.

$$3x+8y+33+6y=78$$

Combine 8y and 6y to get 14y.

$$3x+14y+33=78$$

Subtract 33 from both sides.

$$3x+14y=78-33$$

Subtract 33 from 78 to get 45.

$$3x+14y=45$$

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

$$3x+14y=45,13x-7y=-8$$

Write the equations in matrix form.

$$\left(\begin{matrix}3&14\\13&-7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}45\\-8\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}3&14\\13&-7\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&14\\13&-7\end{matrix}\right))\left(\begin{matrix}3&14\\13&-7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&14\\13&-7\end{matrix}\right))\left(\begin{matrix}45\\-8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&14\\13&-7\end{matrix}\right))\left(\begin{matrix}45\\-8\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&14\\13&-7\end{matrix}\right))\left(\begin{matrix}45\\-8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{3\left(-7\right)-14\times 13}&-\frac{14}{3\left(-7\right)-14\times 13}\\-\frac{13}{3\left(-7\right)-14\times 13}&\frac{3}{3\left(-7\right)-14\times 13}\end{matrix}\right)\left(\begin{matrix}45\\-8\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{29}&\frac{2}{29}\\\frac{13}{203}&-\frac{3}{203}\end{matrix}\right)\left(\begin{matrix}45\\-8\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{29}\times 45+\frac{2}{29}\left(-8\right)\\\frac{13}{203}\times 45-\frac{3}{203}\left(-8\right)\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\3\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=1,y=3$$

Steps Using Elimination

Consider the first equation. Multiply both sides of the equation by 6, the least common multiple of 3,2.

$$2\left(9x+4y\right)-3\left(5x-11\right)=78-6y$$

Use the distributive property to multiply 2 by 9x+4y.

$$18x+8y-3\left(5x-11\right)=78-6y$$

Use the distributive property to multiply -3 by 5x-11.

$$18x+8y-15x+33=78-6y$$

Combine 18x and -15x to get 3x.

$$3x+8y+33=78-6y$$

Add 6y to both sides.

$$3x+8y+33+6y=78$$

Combine 8y and 6y to get 14y.

$$3x+14y+33=78$$

Subtract 33 from both sides.

$$3x+14y=78-33$$

Subtract 33 from 78 to get 45.

$$3x+14y=45$$

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.

$$3x+14y=45,13x-7y=-8$$

To make 3x 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 3.

$$13\times 3x+13\times 14y=13\times 45,3\times 13x+3\left(-7\right)y=3\left(-8\right)$$

Simplify.

$$39x+182y=585,39x-21y=-24$$

Subtract 39x-21y=-24 from 39x+182y=585 by subtracting like terms on each side of the equal sign.

$$39x-39x+182y+21y=585+24$$

Add 39x to -39x. Terms 39x and -39x cancel out, leaving an equation with only one variable that can be solved.

$$182y+21y=585+24$$

Add 182y to 21y.

$$203y=585+24$$

Add 585 to 24.

$$203y=609$$

Divide both sides by 203.

$$y=3$$

Substitute 3 for y in 13x-7y=-8. Because the resulting equation contains only one variable, you can solve for x directly.

$$13x-7\times 3=-8$$

Multiply -7 times 3.

$$13x-21=-8$$

Add 21 to both sides of the equation.

$$13x=13$$

Divide both sides by 13.

$$x=1$$

The system is now solved.

$$x=1,y=3$$