Solve {l}{6(x+y)+8(x-y)=2210}{3(x+y)-4(x-y)=120}

$$\left. \begin{array} { l } { 6 ( x + y ) + 8 ( x - y ) = 2210 } \\ { 3 ( x + y ) - 4 ( x - y ) = 120 } \end{array} \right.$$

Solve for x, y

$$x = \frac{7855}{48} = 163\frac{31}{48} \approx 163.645833333$$

$$y = \frac{1945}{48} = 40\frac{25}{48} \approx 40.520833333$$

Steps Using Substitution

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

$$6x+6y+8\left(x-y\right)=2210$$

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

$$6x+6y+8x-8y=2210$$

Combine 6x and 8x to get 14x.

$$14x+6y-8y=2210$$

Combine 6y and -8y to get -2y.

$$14x-2y=2210$$

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

$$3x+3y-4\left(x-y\right)=120$$

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

$$3x+3y-4x+4y=120$$

Combine 3x and -4x to get -x.

$$-x+3y+4y=120$$

Combine 3y and 4y to get 7y.

$$-x+7y=120$$

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.

$$14x-2y=2210,-x+7y=120$$

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

$$14x-2y=2210$$

Add 2y to both sides of the equation.

$$14x=2y+2210$$

Divide both sides by 14.

$$x=\frac{1}{14}\left(2y+2210\right)$$

Multiply \frac{1}{14} times 2210+2y.

$$x=\frac{1}{7}y+\frac{1105}{7}$$

Substitute \frac{1105+y}{7} for x in the other equation, -x+7y=120.

$$-\left(\frac{1}{7}y+\frac{1105}{7}\right)+7y=120$$

Multiply -1 times \frac{1105+y}{7}.

$$-\frac{1}{7}y-\frac{1105}{7}+7y=120$$

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

$$\frac{48}{7}y-\frac{1105}{7}=120$$

Add \frac{1105}{7} to both sides of the equation.

$$\frac{48}{7}y=\frac{1945}{7}$$

Divide both sides of the equation by \frac{48}{7}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=\frac{1945}{48}$$

Substitute \frac{1945}{48} for y in x=\frac{1}{7}y+\frac{1105}{7}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{1}{7}\times \frac{1945}{48}+\frac{1105}{7}$$

Multiply \frac{1}{7} times \frac{1945}{48} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{1945}{336}+\frac{1105}{7}$$

Add \frac{1105}{7} to \frac{1945}{336} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{7855}{48}$$

The system is now solved.

$$x=\frac{7855}{48},y=\frac{1945}{48}$$

Steps Using Matrices

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

$$6x+6y+8\left(x-y\right)=2210$$

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

$$6x+6y+8x-8y=2210$$

Combine 6x and 8x to get 14x.

$$14x+6y-8y=2210$$

Combine 6y and -8y to get -2y.

$$14x-2y=2210$$

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

$$3x+3y-4\left(x-y\right)=120$$

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

$$3x+3y-4x+4y=120$$

Combine 3x and -4x to get -x.

$$-x+3y+4y=120$$

Combine 3y and 4y to get 7y.

$$-x+7y=120$$

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

$$14x-2y=2210,-x+7y=120$$

Write the equations in matrix form.

$$\left(\begin{matrix}14&-2\\-1&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2210\\120\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}14&-2\\-1&7\end{matrix}\right).

$$inverse(\left(\begin{matrix}14&-2\\-1&7\end{matrix}\right))\left(\begin{matrix}14&-2\\-1&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}14&-2\\-1&7\end{matrix}\right))\left(\begin{matrix}2210\\120\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}14&-2\\-1&7\end{matrix}\right))\left(\begin{matrix}2210\\120\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}14&-2\\-1&7\end{matrix}\right))\left(\begin{matrix}2210\\120\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{14\times 7-\left(-2\left(-1\right)\right)}&-\frac{-2}{14\times 7-\left(-2\left(-1\right)\right)}\\-\frac{-1}{14\times 7-\left(-2\left(-1\right)\right)}&\frac{14}{14\times 7-\left(-2\left(-1\right)\right)}\end{matrix}\right)\left(\begin{matrix}2210\\120\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{96}&\frac{1}{48}\\\frac{1}{96}&\frac{7}{48}\end{matrix}\right)\left(\begin{matrix}2210\\120\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{96}\times 2210+\frac{1}{48}\times 120\\\frac{1}{96}\times 2210+\frac{7}{48}\times 120\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7855}{48}\\\frac{1945}{48}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{7855}{48},y=\frac{1945}{48}$$

Steps Using Elimination

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

$$6x+6y+8\left(x-y\right)=2210$$

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

$$6x+6y+8x-8y=2210$$

Combine 6x and 8x to get 14x.

$$14x+6y-8y=2210$$

Combine 6y and -8y to get -2y.

$$14x-2y=2210$$

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

$$3x+3y-4\left(x-y\right)=120$$

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

$$3x+3y-4x+4y=120$$

Combine 3x and -4x to get -x.

$$-x+3y+4y=120$$

Combine 3y and 4y to get 7y.

$$-x+7y=120$$

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.

$$14x-2y=2210,-x+7y=120$$

To make 14x and -x equal, multiply all terms on each side of the first equation by -1 and all terms on each side of the second by 14.

$$-14x-\left(-2y\right)=-2210,14\left(-1\right)x+14\times 7y=14\times 120$$

Simplify.

$$-14x+2y=-2210,-14x+98y=1680$$

Subtract -14x+98y=1680 from -14x+2y=-2210 by subtracting like terms on each side of the equal sign.

$$-14x+14x+2y-98y=-2210-1680$$

Add -14x to 14x. Terms -14x and 14x cancel out, leaving an equation with only one variable that can be solved.

$$2y-98y=-2210-1680$$

Add 2y to -98y.

$$-96y=-2210-1680$$

Add -2210 to -1680.

$$-96y=-3890$$

Divide both sides by -96.

$$y=\frac{1945}{48}$$

Substitute \frac{1945}{48} for y in -x+7y=120. Because the resulting equation contains only one variable, you can solve for x directly.

$$-x+7\times \frac{1945}{48}=120$$

Multiply 7 times \frac{1945}{48}.

$$-x+\frac{13615}{48}=120$$

Subtract \frac{13615}{48} from both sides of the equation.

$$-x=-\frac{7855}{48}$$

Divide both sides by -1.

$$x=\frac{7855}{48}$$

The system is now solved.

$$x=\frac{7855}{48},y=\frac{1945}{48}$$