Solve {l}{6x-3y+2x-4y+4=0}{x+-8y=-35(-15)x+8y=35}

$$\left. \begin{array} { l } { 6 x - 3 y + 2 x - 4 y + 4 = 0 } \\ { x + - 8 y = - 35 ( - 15 ) x + 8 y = 35 } \end{array} \right.$$

Solve for x, y

$$x=-\frac{16}{949}\approx -0.016859852$$

$$y=\frac{524}{949}\approx 0.552160169$$

Steps Using Substitution

Consider the first equation. Combine 6x and 2x to get 8x.

$$8x-3y-4y+4=0$$

Combine -3y and -4y to get -7y.

$$8x-7y+4=0$$

Subtract 4 from both sides. Anything subtracted from zero gives its negation.

$$8x-7y=-4$$

Consider the second equation. Multiply -35 and -15 to get 525.

$$x-8y=525x+8y$$

Subtract 525x from both sides.

$$x-8y-525x=8y$$

Combine x and -525x to get -524x.

$$-524x-8y=8y$$

Subtract 8y from both sides.

$$-524x-8y-8y=0$$

Combine -8y and -8y to get -16y.

$$-524x-16y=0$$

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.

$$8x-7y=-4,-524x-16y=0$$

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

$$8x-7y=-4$$

Add 7y to both sides of the equation.

$$8x=7y-4$$

Divide both sides by 8.

$$x=\frac{1}{8}\left(7y-4\right)$$

Multiply \frac{1}{8} times 7y-4.

$$x=\frac{7}{8}y-\frac{1}{2}$$

Substitute \frac{7y}{8}-\frac{1}{2} for x in the other equation, -524x-16y=0.

$$-524\left(\frac{7}{8}y-\frac{1}{2}\right)-16y=0$$

Multiply -524 times \frac{7y}{8}-\frac{1}{2}.

$$-\frac{917}{2}y+262-16y=0$$

Add -\frac{917y}{2} to -16y.

$$-\frac{949}{2}y+262=0$$

Subtract 262 from both sides of the equation.

$$-\frac{949}{2}y=-262$$

Divide both sides of the equation by -\frac{949}{2}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=\frac{524}{949}$$

Substitute \frac{524}{949} for y in x=\frac{7}{8}y-\frac{1}{2}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{7}{8}\times \frac{524}{949}-\frac{1}{2}$$

Multiply \frac{7}{8} times \frac{524}{949} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{917}{1898}-\frac{1}{2}$$

Add -\frac{1}{2} to \frac{917}{1898} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{16}{949}$$

The system is now solved.

$$x=-\frac{16}{949},y=\frac{524}{949}$$

Steps Using Matrices

Consider the first equation. Combine 6x and 2x to get 8x.

$$8x-3y-4y+4=0$$

Combine -3y and -4y to get -7y.

$$8x-7y+4=0$$

Subtract 4 from both sides. Anything subtracted from zero gives its negation.

$$8x-7y=-4$$

Consider the second equation. Multiply -35 and -15 to get 525.

$$x-8y=525x+8y$$

Subtract 525x from both sides.

$$x-8y-525x=8y$$

Combine x and -525x to get -524x.

$$-524x-8y=8y$$

Subtract 8y from both sides.

$$-524x-8y-8y=0$$

Combine -8y and -8y to get -16y.

$$-524x-16y=0$$

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

$$8x-7y=-4,-524x-16y=0$$

Write the equations in matrix form.

$$\left(\begin{matrix}8&-7\\-524&-16\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-4\\0\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}8&-7\\-524&-16\end{matrix}\right).

$$inverse(\left(\begin{matrix}8&-7\\-524&-16\end{matrix}\right))\left(\begin{matrix}8&-7\\-524&-16\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&-7\\-524&-16\end{matrix}\right))\left(\begin{matrix}-4\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&-7\\-524&-16\end{matrix}\right))\left(\begin{matrix}-4\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&-7\\-524&-16\end{matrix}\right))\left(\begin{matrix}-4\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{16}{8\left(-16\right)-\left(-7\left(-524\right)\right)}&-\frac{-7}{8\left(-16\right)-\left(-7\left(-524\right)\right)}\\-\frac{-524}{8\left(-16\right)-\left(-7\left(-524\right)\right)}&\frac{8}{8\left(-16\right)-\left(-7\left(-524\right)\right)}\end{matrix}\right)\left(\begin{matrix}-4\\0\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{949}&-\frac{7}{3796}\\-\frac{131}{949}&-\frac{2}{949}\end{matrix}\right)\left(\begin{matrix}-4\\0\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{949}\left(-4\right)\\-\frac{131}{949}\left(-4\right)\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{16}{949}\\\frac{524}{949}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=-\frac{16}{949},y=\frac{524}{949}$$

Steps Using Elimination

Consider the first equation. Combine 6x and 2x to get 8x.

$$8x-3y-4y+4=0$$

Combine -3y and -4y to get -7y.

$$8x-7y+4=0$$

Subtract 4 from both sides. Anything subtracted from zero gives its negation.

$$8x-7y=-4$$

Consider the second equation. Multiply -35 and -15 to get 525.

$$x-8y=525x+8y$$

Subtract 525x from both sides.

$$x-8y-525x=8y$$

Combine x and -525x to get -524x.

$$-524x-8y=8y$$

Subtract 8y from both sides.

$$-524x-8y-8y=0$$

Combine -8y and -8y to get -16y.

$$-524x-16y=0$$

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.

$$8x-7y=-4,-524x-16y=0$$

To make 8x and -524x equal, multiply all terms on each side of the first equation by -524 and all terms on each side of the second by 8.

$$-524\times 8x-524\left(-7\right)y=-524\left(-4\right),8\left(-524\right)x+8\left(-16\right)y=0$$

Simplify.

$$-4192x+3668y=2096,-4192x-128y=0$$

Subtract -4192x-128y=0 from -4192x+3668y=2096 by subtracting like terms on each side of the equal sign.

$$-4192x+4192x+3668y+128y=2096$$

Add -4192x to 4192x. Terms -4192x and 4192x cancel out, leaving an equation with only one variable that can be solved.

$$3668y+128y=2096$$

Add 3668y to 128y.

$$3796y=2096$$

Divide both sides by 3796.

$$y=\frac{524}{949}$$

Substitute \frac{524}{949} for y in -524x-16y=0. Because the resulting equation contains only one variable, you can solve for x directly.

$$-524x-16\times \frac{524}{949}=0$$

Multiply -16 times \frac{524}{949}.

$$-524x-\frac{8384}{949}=0$$

Add \frac{8384}{949} to both sides of the equation.

$$-524x=\frac{8384}{949}$$

Divide both sides by -524.

$$x=-\frac{16}{949}$$

The system is now solved.

$$x=-\frac{16}{949},y=\frac{524}{949}$$