Solve {l}{2x-y+3/3-x-2y+3/4=4}{3x-4y+3/4+4x-2y-9/3=4}

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

Solve for x, y

$$x=7$$

$$y=5$$

Steps Using Substitution

Consider the first equation. Multiply both sides of the equation by 12, the least common multiple of 3,4.

$$4\left(2x-y+3\right)-3\left(x-2y+3\right)=48$$

Use the distributive property to multiply 4 by 2x-y+3.

$$8x-4y+12-3\left(x-2y+3\right)=48$$

Use the distributive property to multiply -3 by x-2y+3.

$$8x-4y+12-3x+6y-9=48$$

Combine 8x and -3x to get 5x.

$$5x-4y+12+6y-9=48$$

Combine -4y and 6y to get 2y.

$$5x+2y+12-9=48$$

Subtract 9 from 12 to get 3.

$$5x+2y+3=48$$

Subtract 3 from both sides.

$$5x+2y=48-3$$

Subtract 3 from 48 to get 45.

$$5x+2y=45$$

Consider the second equation. Multiply both sides of the equation by 12, the least common multiple of 4,3.

$$3\left(3x-4y+3\right)+4\left(4x-2y-9\right)=48$$

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

$$9x-12y+9+4\left(4x-2y-9\right)=48$$

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

$$9x-12y+9+16x-8y-36=48$$

Combine 9x and 16x to get 25x.

$$25x-12y+9-8y-36=48$$

Combine -12y and -8y to get -20y.

$$25x-20y+9-36=48$$

Subtract 36 from 9 to get -27.

$$25x-20y-27=48$$

Add 27 to both sides.

$$25x-20y=48+27$$

Add 48 and 27 to get 75.

$$25x-20y=75$$

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.

$$5x+2y=45,25x-20y=75$$

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

$$5x+2y=45$$

Subtract 2y from both sides of the equation.

$$5x=-2y+45$$

Divide both sides by 5.

$$x=\frac{1}{5}\left(-2y+45\right)$$

Multiply \frac{1}{5} times -2y+45.

$$x=-\frac{2}{5}y+9$$

Substitute -\frac{2y}{5}+9 for x in the other equation, 25x-20y=75.

$$25\left(-\frac{2}{5}y+9\right)-20y=75$$

Multiply 25 times -\frac{2y}{5}+9.

$$-10y+225-20y=75$$

Add -10y to -20y.

$$-30y+225=75$$

Subtract 225 from both sides of the equation.

$$-30y=-150$$

Divide both sides by -30.

$$y=5$$

Substitute 5 for y in x=-\frac{2}{5}y+9. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{2}{5}\times 5+9$$

Multiply -\frac{2}{5} times 5.

$$x=-2+9$$

Add 9 to -2.

$$x=7$$

The system is now solved.

$$x=7,y=5$$

Steps Using Matrices

Consider the first equation. Multiply both sides of the equation by 12, the least common multiple of 3,4.

$$4\left(2x-y+3\right)-3\left(x-2y+3\right)=48$$

Use the distributive property to multiply 4 by 2x-y+3.

$$8x-4y+12-3\left(x-2y+3\right)=48$$

Use the distributive property to multiply -3 by x-2y+3.

$$8x-4y+12-3x+6y-9=48$$

Combine 8x and -3x to get 5x.

$$5x-4y+12+6y-9=48$$

Combine -4y and 6y to get 2y.

$$5x+2y+12-9=48$$

Subtract 9 from 12 to get 3.

$$5x+2y+3=48$$

Subtract 3 from both sides.

$$5x+2y=48-3$$

Subtract 3 from 48 to get 45.

$$5x+2y=45$$

Consider the second equation. Multiply both sides of the equation by 12, the least common multiple of 4,3.

$$3\left(3x-4y+3\right)+4\left(4x-2y-9\right)=48$$

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

$$9x-12y+9+4\left(4x-2y-9\right)=48$$

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

$$9x-12y+9+16x-8y-36=48$$

Combine 9x and 16x to get 25x.

$$25x-12y+9-8y-36=48$$

Combine -12y and -8y to get -20y.

$$25x-20y+9-36=48$$

Subtract 36 from 9 to get -27.

$$25x-20y-27=48$$

Add 27 to both sides.

$$25x-20y=48+27$$

Add 48 and 27 to get 75.

$$25x-20y=75$$

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

$$5x+2y=45,25x-20y=75$$

Write the equations in matrix form.

$$\left(\begin{matrix}5&2\\25&-20\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}45\\75\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}5&2\\25&-20\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&2\\25&-20\end{matrix}\right))\left(\begin{matrix}5&2\\25&-20\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&2\\25&-20\end{matrix}\right))\left(\begin{matrix}45\\75\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&2\\25&-20\end{matrix}\right))\left(\begin{matrix}45\\75\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&2\\25&-20\end{matrix}\right))\left(\begin{matrix}45\\75\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{20}{5\left(-20\right)-2\times 25}&-\frac{2}{5\left(-20\right)-2\times 25}\\-\frac{25}{5\left(-20\right)-2\times 25}&\frac{5}{5\left(-20\right)-2\times 25}\end{matrix}\right)\left(\begin{matrix}45\\75\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{15}&\frac{1}{75}\\\frac{1}{6}&-\frac{1}{30}\end{matrix}\right)\left(\begin{matrix}45\\75\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{15}\times 45+\frac{1}{75}\times 75\\\frac{1}{6}\times 45-\frac{1}{30}\times 75\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\5\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=7,y=5$$

Steps Using Elimination

Consider the first equation. Multiply both sides of the equation by 12, the least common multiple of 3,4.

$$4\left(2x-y+3\right)-3\left(x-2y+3\right)=48$$

Use the distributive property to multiply 4 by 2x-y+3.

$$8x-4y+12-3\left(x-2y+3\right)=48$$

Use the distributive property to multiply -3 by x-2y+3.

$$8x-4y+12-3x+6y-9=48$$

Combine 8x and -3x to get 5x.

$$5x-4y+12+6y-9=48$$

Combine -4y and 6y to get 2y.

$$5x+2y+12-9=48$$

Subtract 9 from 12 to get 3.

$$5x+2y+3=48$$

Subtract 3 from both sides.

$$5x+2y=48-3$$

Subtract 3 from 48 to get 45.

$$5x+2y=45$$

Consider the second equation. Multiply both sides of the equation by 12, the least common multiple of 4,3.

$$3\left(3x-4y+3\right)+4\left(4x-2y-9\right)=48$$

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

$$9x-12y+9+4\left(4x-2y-9\right)=48$$

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

$$9x-12y+9+16x-8y-36=48$$

Combine 9x and 16x to get 25x.

$$25x-12y+9-8y-36=48$$

Combine -12y and -8y to get -20y.

$$25x-20y+9-36=48$$

Subtract 36 from 9 to get -27.

$$25x-20y-27=48$$

Add 27 to both sides.

$$25x-20y=48+27$$

Add 48 and 27 to get 75.

$$25x-20y=75$$

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.

$$5x+2y=45,25x-20y=75$$

To make 5x and 25x equal, multiply all terms on each side of the first equation by 25 and all terms on each side of the second by 5.

$$25\times 5x+25\times 2y=25\times 45,5\times 25x+5\left(-20\right)y=5\times 75$$

Simplify.

$$125x+50y=1125,125x-100y=375$$

Subtract 125x-100y=375 from 125x+50y=1125 by subtracting like terms on each side of the equal sign.

$$125x-125x+50y+100y=1125-375$$

Add 125x to -125x. Terms 125x and -125x cancel out, leaving an equation with only one variable that can be solved.

$$50y+100y=1125-375$$

Add 50y to 100y.

$$150y=1125-375$$

Add 1125 to -375.

$$150y=750$$

Divide both sides by 150.

$$y=5$$

Substitute 5 for y in 25x-20y=75. Because the resulting equation contains only one variable, you can solve for x directly.

$$25x-20\times 5=75$$

Multiply -20 times 5.

$$25x-100=75$$

Add 100 to both sides of the equation.

$$25x=175$$

Divide both sides by 25.

$$x=7$$

The system is now solved.

$$x=7,y=5$$