Solve {r}{6(x-2y)-3(x+y)=72}{5(x-3y)-4(3x+2y)=64}

$$\left. \begin{array} { r } { 6 ( x - 2 y ) - 3 ( x + y ) = 72 } \\ { 5 ( x - 3 y ) - 4 ( 3 x + 2 y ) = 64 } \end{array} \right.$$

Solve for x, y

$$x=4$$

$$y=-4$$

Steps Using Substitution

Consider the first equation. Use the distributive property to multiply 6 by x-2y.

$$6x-12y-3\left(x+y\right)=72$$

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

$$6x-12y-3x-3y=72$$

Combine 6x and -3x to get 3x.

$$3x-12y-3y=72$$

Combine -12y and -3y to get -15y.

$$3x-15y=72$$

Consider the second equation. Use the distributive property to multiply 5 by x-3y.

$$5x-15y-4\left(3x+2y\right)=64$$

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

$$5x-15y-12x-8y=64$$

Combine 5x and -12x to get -7x.

$$-7x-15y-8y=64$$

Combine -15y and -8y to get -23y.

$$-7x-23y=64$$

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.

$$3x-15y=72,-7x-23y=64$$

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

$$3x-15y=72$$

Add 15y to both sides of the equation.

$$3x=15y+72$$

Divide both sides by 3.

$$x=\frac{1}{3}\left(15y+72\right)$$

Multiply \frac{1}{3} times 15y+72.

$$x=5y+24$$

Substitute 5y+24 for x in the other equation, -7x-23y=64.

$$-7\left(5y+24\right)-23y=64$$

Multiply -7 times 5y+24.

$$-35y-168-23y=64$$

Add -35y to -23y.

$$-58y-168=64$$

Add 168 to both sides of the equation.

$$-58y=232$$

Divide both sides by -58.

$$y=-4$$

Substitute -4 for y in x=5y+24. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=5\left(-4\right)+24$$

Multiply 5 times -4.

$$x=-20+24$$

Add 24 to -20.

$$x=4$$

The system is now solved.

$$x=4,y=-4$$

Steps Using Matrices

Consider the first equation. Use the distributive property to multiply 6 by x-2y.

$$6x-12y-3\left(x+y\right)=72$$

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

$$6x-12y-3x-3y=72$$

Combine 6x and -3x to get 3x.

$$3x-12y-3y=72$$

Combine -12y and -3y to get -15y.

$$3x-15y=72$$

Consider the second equation. Use the distributive property to multiply 5 by x-3y.

$$5x-15y-4\left(3x+2y\right)=64$$

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

$$5x-15y-12x-8y=64$$

Combine 5x and -12x to get -7x.

$$-7x-15y-8y=64$$

Combine -15y and -8y to get -23y.

$$-7x-23y=64$$

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

$$3x-15y=72,-7x-23y=64$$

Write the equations in matrix form.

$$\left(\begin{matrix}3&-15\\-7&-23\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}72\\64\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}3&-15\\-7&-23\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-15\\-7&-23\end{matrix}\right))\left(\begin{matrix}3&-15\\-7&-23\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-15\\-7&-23\end{matrix}\right))\left(\begin{matrix}72\\64\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-15\\-7&-23\end{matrix}\right))\left(\begin{matrix}72\\64\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-15\\-7&-23\end{matrix}\right))\left(\begin{matrix}72\\64\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{23}{3\left(-23\right)-\left(-15\left(-7\right)\right)}&-\frac{-15}{3\left(-23\right)-\left(-15\left(-7\right)\right)}\\-\frac{-7}{3\left(-23\right)-\left(-15\left(-7\right)\right)}&\frac{3}{3\left(-23\right)-\left(-15\left(-7\right)\right)}\end{matrix}\right)\left(\begin{matrix}72\\64\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{23}{174}&-\frac{5}{58}\\-\frac{7}{174}&-\frac{1}{58}\end{matrix}\right)\left(\begin{matrix}72\\64\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{23}{174}\times 72-\frac{5}{58}\times 64\\-\frac{7}{174}\times 72-\frac{1}{58}\times 64\end{matrix}\right)$$

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

Consider the first equation. Use the distributive property to multiply 6 by x-2y.

$$6x-12y-3\left(x+y\right)=72$$

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

$$6x-12y-3x-3y=72$$

Combine 6x and -3x to get 3x.

$$3x-12y-3y=72$$

Combine -12y and -3y to get -15y.

$$3x-15y=72$$

Consider the second equation. Use the distributive property to multiply 5 by x-3y.

$$5x-15y-4\left(3x+2y\right)=64$$

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

$$5x-15y-12x-8y=64$$

Combine 5x and -12x to get -7x.

$$-7x-15y-8y=64$$

Combine -15y and -8y to get -23y.

$$-7x-23y=64$$

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.

$$3x-15y=72,-7x-23y=64$$

To make 3x and -7x equal, multiply all terms on each side of the first equation by -7 and all terms on each side of the second by 3.

$$-7\times 3x-7\left(-15\right)y=-7\times 72,3\left(-7\right)x+3\left(-23\right)y=3\times 64$$

Simplify.

$$-21x+105y=-504,-21x-69y=192$$

Subtract -21x-69y=192 from -21x+105y=-504 by subtracting like terms on each side of the equal sign.

$$-21x+21x+105y+69y=-504-192$$

Add -21x to 21x. Terms -21x and 21x cancel out, leaving an equation with only one variable that can be solved.

$$105y+69y=-504-192$$

Add 105y to 69y.

$$174y=-504-192$$

Add -504 to -192.

$$174y=-696$$

Divide both sides by 174.

$$y=-4$$

Substitute -4 for y in -7x-23y=64. Because the resulting equation contains only one variable, you can solve for x directly.

$$-7x-23\left(-4\right)=64$$

Multiply -23 times -4.

$$-7x+92=64$$

Subtract 92 from both sides of the equation.

$$-7x=-28$$

Divide both sides by -7.

$$x=4$$

The system is now solved.

$$x=4,y=-4$$