Solve {l}{5(3x+y)-8(x-6y)=20}{6(x-10y)-13(x-y)=52}

$$\left\{ \begin{array} { l } { 5 ( 3 x + y ) - 8 ( x - 6 y ) = 20 } \\ { 6 ( x - 10 y ) - 13 ( x - y ) = 52 } \end{array} \right.$$

Solve for x, y

$$x=-88$$

$$y=12$$

Steps Using Substitution

Consider the first equation. Use the distributive property to multiply 5 by 3x+y.

$$15x+5y-8\left(x-6y\right)=20$$

Use the distributive property to multiply -8 by x-6y.

$$15x+5y-8x+48y=20$$

Combine 15x and -8x to get 7x.

$$7x+5y+48y=20$$

Combine 5y and 48y to get 53y.

$$7x+53y=20$$

Consider the second equation. Use the distributive property to multiply 6 by x-10y.

$$6x-60y-13\left(x-y\right)=52$$

Use the distributive property to multiply -13 by x-y.

$$6x-60y-13x+13y=52$$

Combine 6x and -13x to get -7x.

$$-7x-60y+13y=52$$

Combine -60y and 13y to get -47y.

$$-7x-47y=52$$

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.

$$7x+53y=20,-7x-47y=52$$

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

$$7x+53y=20$$

Subtract 53y from both sides of the equation.

$$7x=-53y+20$$

Divide both sides by 7.

$$x=\frac{1}{7}\left(-53y+20\right)$$

Multiply \frac{1}{7} times -53y+20.

$$x=-\frac{53}{7}y+\frac{20}{7}$$

Substitute \frac{-53y+20}{7} for x in the other equation, -7x-47y=52.

$$-7\left(-\frac{53}{7}y+\frac{20}{7}\right)-47y=52$$

Multiply -7 times \frac{-53y+20}{7}.

$$53y-20-47y=52$$

Add 53y to -47y.

$$6y-20=52$$

Add 20 to both sides of the equation.

$$6y=72$$

Divide both sides by 6.

$$y=12$$

Substitute 12 for y in x=-\frac{53}{7}y+\frac{20}{7}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{53}{7}\times 12+\frac{20}{7}$$

Multiply -\frac{53}{7} times 12.

$$x=\frac{-636+20}{7}$$

Add \frac{20}{7} to -\frac{636}{7} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-88$$

The system is now solved.

$$x=-88,y=12$$

Steps Using Matrices

Consider the first equation. Use the distributive property to multiply 5 by 3x+y.

$$15x+5y-8\left(x-6y\right)=20$$

Use the distributive property to multiply -8 by x-6y.

$$15x+5y-8x+48y=20$$

Combine 15x and -8x to get 7x.

$$7x+5y+48y=20$$

Combine 5y and 48y to get 53y.

$$7x+53y=20$$

Consider the second equation. Use the distributive property to multiply 6 by x-10y.

$$6x-60y-13\left(x-y\right)=52$$

Use the distributive property to multiply -13 by x-y.

$$6x-60y-13x+13y=52$$

Combine 6x and -13x to get -7x.

$$-7x-60y+13y=52$$

Combine -60y and 13y to get -47y.

$$-7x-47y=52$$

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

$$7x+53y=20,-7x-47y=52$$

Write the equations in matrix form.

$$\left(\begin{matrix}7&53\\-7&-47\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}20\\52\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}7&53\\-7&-47\end{matrix}\right).

$$inverse(\left(\begin{matrix}7&53\\-7&-47\end{matrix}\right))\left(\begin{matrix}7&53\\-7&-47\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&53\\-7&-47\end{matrix}\right))\left(\begin{matrix}20\\52\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&53\\-7&-47\end{matrix}\right))\left(\begin{matrix}20\\52\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&53\\-7&-47\end{matrix}\right))\left(\begin{matrix}20\\52\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{47}{7\left(-47\right)-53\left(-7\right)}&-\frac{53}{7\left(-47\right)-53\left(-7\right)}\\-\frac{-7}{7\left(-47\right)-53\left(-7\right)}&\frac{7}{7\left(-47\right)-53\left(-7\right)}\end{matrix}\right)\left(\begin{matrix}20\\52\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{47}{42}&-\frac{53}{42}\\\frac{1}{6}&\frac{1}{6}\end{matrix}\right)\left(\begin{matrix}20\\52\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{47}{42}\times 20-\frac{53}{42}\times 52\\\frac{1}{6}\times 20+\frac{1}{6}\times 52\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-88\\12\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=-88,y=12$$

Steps Using Elimination

Consider the first equation. Use the distributive property to multiply 5 by 3x+y.

$$15x+5y-8\left(x-6y\right)=20$$

Use the distributive property to multiply -8 by x-6y.

$$15x+5y-8x+48y=20$$

Combine 15x and -8x to get 7x.

$$7x+5y+48y=20$$

Combine 5y and 48y to get 53y.

$$7x+53y=20$$

Consider the second equation. Use the distributive property to multiply 6 by x-10y.

$$6x-60y-13\left(x-y\right)=52$$

Use the distributive property to multiply -13 by x-y.

$$6x-60y-13x+13y=52$$

Combine 6x and -13x to get -7x.

$$-7x-60y+13y=52$$

Combine -60y and 13y to get -47y.

$$-7x-47y=52$$

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.

$$7x+53y=20,-7x-47y=52$$

To make 7x 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 7.

$$-7\times 7x-7\times 53y=-7\times 20,7\left(-7\right)x+7\left(-47\right)y=7\times 52$$

Simplify.

$$-49x-371y=-140,-49x-329y=364$$

Subtract -49x-329y=364 from -49x-371y=-140 by subtracting like terms on each side of the equal sign.

$$-49x+49x-371y+329y=-140-364$$

Add -49x to 49x. Terms -49x and 49x cancel out, leaving an equation with only one variable that can be solved.

$$-371y+329y=-140-364$$

Add -371y to 329y.

$$-42y=-140-364$$

Add -140 to -364.

$$-42y=-504$$

Divide both sides by -42.

$$y=12$$

Substitute 12 for y in -7x-47y=52. Because the resulting equation contains only one variable, you can solve for x directly.

$$-7x-47\times 12=52$$

Multiply -47 times 12.

$$-7x-564=52$$

Add 564 to both sides of the equation.

$$-7x=616$$

Divide both sides by -7.

$$x=-88$$

The system is now solved.

$$x=-88,y=12$$