Solve {l}{5(x-3)/4-3(2y+1)/10=4-7(x+y+1)/8}{6x-5(2y-7)=21}

$$\left\{ \begin{array} { l } { \frac { 5 ( x - 3 ) } { 4 } - \frac { 3 ( 2 y + 1 ) } { 10 } = \frac { 4 - 7 ( x + y + 1 ) } { 8 } } \\ { 6 x - 5 ( 2 y - 7 ) = 21 } \end{array} \right.$$

Solve for x, y

$$x = \frac{329}{229} = 1\frac{100}{229} \approx 1.436681223$$

$$y = \frac{518}{229} = 2\frac{60}{229} \approx 2.262008734$$

Steps Using Substitution

Consider the first equation. Multiply both sides of the equation by 40, the least common multiple of 4,10,8.

$$10\times 5\left(x-3\right)-4\times 3\left(2y+1\right)=5\left(4-7\left(x+y+1\right)\right)$$

Multiply 10 and 5 to get 50.

$$50\left(x-3\right)-4\times 3\left(2y+1\right)=5\left(4-7\left(x+y+1\right)\right)$$

Use the distributive property to multiply 50 by x-3.

$$50x-150-4\times 3\left(2y+1\right)=5\left(4-7\left(x+y+1\right)\right)$$

Multiply -4 and 3 to get -12.

$$50x-150-12\left(2y+1\right)=5\left(4-7\left(x+y+1\right)\right)$$

Use the distributive property to multiply -12 by 2y+1.

$$50x-150-24y-12=5\left(4-7\left(x+y+1\right)\right)$$

Subtract 12 from -150 to get -162.

$$50x-162-24y=5\left(4-7\left(x+y+1\right)\right)$$

Use the distributive property to multiply -7 by x+y+1.

$$50x-162-24y=5\left(4-7x-7y-7\right)$$

Subtract 7 from 4 to get -3.

$$50x-162-24y=5\left(-3-7x-7y\right)$$

Use the distributive property to multiply 5 by -3-7x-7y.

$$50x-162-24y=-15-35x-35y$$

Add 35x to both sides.

$$50x-162-24y+35x=-15-35y$$

Combine 50x and 35x to get 85x.

$$85x-162-24y=-15-35y$$

Add 35y to both sides.

$$85x-162-24y+35y=-15$$

Combine -24y and 35y to get 11y.

$$85x-162+11y=-15$$

Add 162 to both sides.

$$85x+11y=-15+162$$

Add -15 and 162 to get 147.

$$85x+11y=147$$

Consider the second equation. Use the distributive property to multiply -5 by 2y-7.

$$6x-10y+35=21$$

Subtract 35 from both sides.

$$6x-10y=21-35$$

Subtract 35 from 21 to get -14.

$$6x-10y=-14$$

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.

$$85x+11y=147,6x-10y=-14$$

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

$$85x+11y=147$$

Subtract 11y from both sides of the equation.

$$85x=-11y+147$$

Divide both sides by 85.

$$x=\frac{1}{85}\left(-11y+147\right)$$

Multiply \frac{1}{85} times -11y+147.

$$x=-\frac{11}{85}y+\frac{147}{85}$$

Substitute \frac{-11y+147}{85} for x in the other equation, 6x-10y=-14.

$$6\left(-\frac{11}{85}y+\frac{147}{85}\right)-10y=-14$$

Multiply 6 times \frac{-11y+147}{85}.

$$-\frac{66}{85}y+\frac{882}{85}-10y=-14$$

Add -\frac{66y}{85} to -10y.

$$-\frac{916}{85}y+\frac{882}{85}=-14$$

Subtract \frac{882}{85} from both sides of the equation.

$$-\frac{916}{85}y=-\frac{2072}{85}$$

Divide both sides of the equation by -\frac{916}{85}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=\frac{518}{229}$$

Substitute \frac{518}{229} for y in x=-\frac{11}{85}y+\frac{147}{85}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{11}{85}\times \frac{518}{229}+\frac{147}{85}$$

Multiply -\frac{11}{85} times \frac{518}{229} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{5698}{19465}+\frac{147}{85}$$

Add \frac{147}{85} to -\frac{5698}{19465} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{329}{229}$$

The system is now solved.

$$x=\frac{329}{229},y=\frac{518}{229}$$

Steps Using Matrices

Consider the first equation. Multiply both sides of the equation by 40, the least common multiple of 4,10,8.

$$10\times 5\left(x-3\right)-4\times 3\left(2y+1\right)=5\left(4-7\left(x+y+1\right)\right)$$

Multiply 10 and 5 to get 50.

$$50\left(x-3\right)-4\times 3\left(2y+1\right)=5\left(4-7\left(x+y+1\right)\right)$$

Use the distributive property to multiply 50 by x-3.

$$50x-150-4\times 3\left(2y+1\right)=5\left(4-7\left(x+y+1\right)\right)$$

Multiply -4 and 3 to get -12.

$$50x-150-12\left(2y+1\right)=5\left(4-7\left(x+y+1\right)\right)$$

Use the distributive property to multiply -12 by 2y+1.

$$50x-150-24y-12=5\left(4-7\left(x+y+1\right)\right)$$

Subtract 12 from -150 to get -162.

$$50x-162-24y=5\left(4-7\left(x+y+1\right)\right)$$

Use the distributive property to multiply -7 by x+y+1.

$$50x-162-24y=5\left(4-7x-7y-7\right)$$

Subtract 7 from 4 to get -3.

$$50x-162-24y=5\left(-3-7x-7y\right)$$

Use the distributive property to multiply 5 by -3-7x-7y.

$$50x-162-24y=-15-35x-35y$$

Add 35x to both sides.

$$50x-162-24y+35x=-15-35y$$

Combine 50x and 35x to get 85x.

$$85x-162-24y=-15-35y$$

Add 35y to both sides.

$$85x-162-24y+35y=-15$$

Combine -24y and 35y to get 11y.

$$85x-162+11y=-15$$

Add 162 to both sides.

$$85x+11y=-15+162$$

Add -15 and 162 to get 147.

$$85x+11y=147$$

Consider the second equation. Use the distributive property to multiply -5 by 2y-7.

$$6x-10y+35=21$$

Subtract 35 from both sides.

$$6x-10y=21-35$$

Subtract 35 from 21 to get -14.

$$6x-10y=-14$$

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

$$85x+11y=147,6x-10y=-14$$

Write the equations in matrix form.

$$\left(\begin{matrix}85&11\\6&-10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}147\\-14\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}85&11\\6&-10\end{matrix}\right).

$$inverse(\left(\begin{matrix}85&11\\6&-10\end{matrix}\right))\left(\begin{matrix}85&11\\6&-10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}85&11\\6&-10\end{matrix}\right))\left(\begin{matrix}147\\-14\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}85&11\\6&-10\end{matrix}\right))\left(\begin{matrix}147\\-14\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}85&11\\6&-10\end{matrix}\right))\left(\begin{matrix}147\\-14\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{10}{85\left(-10\right)-11\times 6}&-\frac{11}{85\left(-10\right)-11\times 6}\\-\frac{6}{85\left(-10\right)-11\times 6}&\frac{85}{85\left(-10\right)-11\times 6}\end{matrix}\right)\left(\begin{matrix}147\\-14\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{458}&\frac{11}{916}\\\frac{3}{458}&-\frac{85}{916}\end{matrix}\right)\left(\begin{matrix}147\\-14\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{458}\times 147+\frac{11}{916}\left(-14\right)\\\frac{3}{458}\times 147-\frac{85}{916}\left(-14\right)\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{329}{229}\\\frac{518}{229}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{329}{229},y=\frac{518}{229}$$

Steps Using Elimination

Consider the first equation. Multiply both sides of the equation by 40, the least common multiple of 4,10,8.

$$10\times 5\left(x-3\right)-4\times 3\left(2y+1\right)=5\left(4-7\left(x+y+1\right)\right)$$

Multiply 10 and 5 to get 50.

$$50\left(x-3\right)-4\times 3\left(2y+1\right)=5\left(4-7\left(x+y+1\right)\right)$$

Use the distributive property to multiply 50 by x-3.

$$50x-150-4\times 3\left(2y+1\right)=5\left(4-7\left(x+y+1\right)\right)$$

Multiply -4 and 3 to get -12.

$$50x-150-12\left(2y+1\right)=5\left(4-7\left(x+y+1\right)\right)$$

Use the distributive property to multiply -12 by 2y+1.

$$50x-150-24y-12=5\left(4-7\left(x+y+1\right)\right)$$

Subtract 12 from -150 to get -162.

$$50x-162-24y=5\left(4-7\left(x+y+1\right)\right)$$

Use the distributive property to multiply -7 by x+y+1.

$$50x-162-24y=5\left(4-7x-7y-7\right)$$

Subtract 7 from 4 to get -3.

$$50x-162-24y=5\left(-3-7x-7y\right)$$

Use the distributive property to multiply 5 by -3-7x-7y.

$$50x-162-24y=-15-35x-35y$$

Add 35x to both sides.

$$50x-162-24y+35x=-15-35y$$

Combine 50x and 35x to get 85x.

$$85x-162-24y=-15-35y$$

Add 35y to both sides.

$$85x-162-24y+35y=-15$$

Combine -24y and 35y to get 11y.

$$85x-162+11y=-15$$

Add 162 to both sides.

$$85x+11y=-15+162$$

Add -15 and 162 to get 147.

$$85x+11y=147$$

Consider the second equation. Use the distributive property to multiply -5 by 2y-7.

$$6x-10y+35=21$$

Subtract 35 from both sides.

$$6x-10y=21-35$$

Subtract 35 from 21 to get -14.

$$6x-10y=-14$$

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.

$$85x+11y=147,6x-10y=-14$$

To make 85x and 6x equal, multiply all terms on each side of the first equation by 6 and all terms on each side of the second by 85.

$$6\times 85x+6\times 11y=6\times 147,85\times 6x+85\left(-10\right)y=85\left(-14\right)$$

Simplify.

$$510x+66y=882,510x-850y=-1190$$

Subtract 510x-850y=-1190 from 510x+66y=882 by subtracting like terms on each side of the equal sign.

$$510x-510x+66y+850y=882+1190$$

Add 510x to -510x. Terms 510x and -510x cancel out, leaving an equation with only one variable that can be solved.

$$66y+850y=882+1190$$

Add 66y to 850y.

$$916y=882+1190$$

Add 882 to 1190.

$$916y=2072$$

Divide both sides by 916.

$$y=\frac{518}{229}$$

Substitute \frac{518}{229} for y in 6x-10y=-14. Because the resulting equation contains only one variable, you can solve for x directly.

$$6x-10\times \frac{518}{229}=-14$$

Multiply -10 times \frac{518}{229}.

$$6x-\frac{5180}{229}=-14$$

Add \frac{5180}{229} to both sides of the equation.

$$6x=\frac{1974}{229}$$

Divide both sides by 6.

$$x=\frac{329}{229}$$

The system is now solved.

$$x=\frac{329}{229},y=\frac{518}{229}$$