Solve {l}{5(x-3)/4-3(2y+1)/10=6-4(x+y-1)/8}{6x-5(2y-7)=48}

$$\left\{ \begin{array} { l } { \frac { 5 ( x - 3 ) } { 4 } - \frac { 3 ( 2 y + 1 ) } { 10 } = \frac { 6 - 4 ( x + y - 1 ) } { 8 } } \\ { 6 x - 5 ( 2 y - 7 ) = 48 } \end{array} \right.$$

Solve for x, y

$$x = \frac{517}{169} = 3\frac{10}{169} \approx 3.059171598$$

$$y=\frac{181}{338}\approx 0.535502959$$

Steps Using Substitution

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

$$10\times 5\left(x-3\right)-4\times 3\left(2y+1\right)=5\left(6-4\left(x+y-1\right)\right)$$

Multiply 10 and 5 to get 50.

$$50\left(x-3\right)-4\times 3\left(2y+1\right)=5\left(6-4\left(x+y-1\right)\right)$$

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

$$50x-150-4\times 3\left(2y+1\right)=5\left(6-4\left(x+y-1\right)\right)$$

Multiply -4 and 3 to get -12.

$$50x-150-12\left(2y+1\right)=5\left(6-4\left(x+y-1\right)\right)$$

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

$$50x-150-24y-12=5\left(6-4\left(x+y-1\right)\right)$$

Subtract 12 from -150 to get -162.

$$50x-162-24y=5\left(6-4\left(x+y-1\right)\right)$$

Use the distributive property to multiply -4 by x+y-1.

$$50x-162-24y=5\left(6-4x-4y+4\right)$$

Add 6 and 4 to get 10.

$$50x-162-24y=5\left(10-4x-4y\right)$$

Use the distributive property to multiply 5 by 10-4x-4y.

$$50x-162-24y=50-20x-20y$$

Add 20x to both sides.

$$50x-162-24y+20x=50-20y$$

Combine 50x and 20x to get 70x.

$$70x-162-24y=50-20y$$

Add 20y to both sides.

$$70x-162-24y+20y=50$$

Combine -24y and 20y to get -4y.

$$70x-162-4y=50$$

Add 162 to both sides.

$$70x-4y=50+162$$

Add 50 and 162 to get 212.

$$70x-4y=212$$

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

$$6x-10y+35=48$$

Subtract 35 from both sides.

$$6x-10y=48-35$$

Subtract 35 from 48 to get 13.

$$6x-10y=13$$

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.

$$70x-4y=212,6x-10y=13$$

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

$$70x-4y=212$$

Add 4y to both sides of the equation.

$$70x=4y+212$$

Divide both sides by 70.

$$x=\frac{1}{70}\left(4y+212\right)$$

Multiply \frac{1}{70} times 212+4y.

$$x=\frac{2}{35}y+\frac{106}{35}$$

Substitute \frac{106+2y}{35} for x in the other equation, 6x-10y=13.

$$6\left(\frac{2}{35}y+\frac{106}{35}\right)-10y=13$$

Multiply 6 times \frac{106+2y}{35}.

$$\frac{12}{35}y+\frac{636}{35}-10y=13$$

Add \frac{12y}{35} to -10y.

$$-\frac{338}{35}y+\frac{636}{35}=13$$

Subtract \frac{636}{35} from both sides of the equation.

$$-\frac{338}{35}y=-\frac{181}{35}$$

Divide both sides of the equation by -\frac{338}{35}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=\frac{181}{338}$$

Substitute \frac{181}{338} for y in x=\frac{2}{35}y+\frac{106}{35}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{2}{35}\times \frac{181}{338}+\frac{106}{35}$$

Multiply \frac{2}{35} times \frac{181}{338} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{181}{5915}+\frac{106}{35}$$

Add \frac{106}{35} to \frac{181}{5915} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{517}{169}$$

The system is now solved.

$$x=\frac{517}{169},y=\frac{181}{338}$$

Steps Using Matrices

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

$$10\times 5\left(x-3\right)-4\times 3\left(2y+1\right)=5\left(6-4\left(x+y-1\right)\right)$$

Multiply 10 and 5 to get 50.

$$50\left(x-3\right)-4\times 3\left(2y+1\right)=5\left(6-4\left(x+y-1\right)\right)$$

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

$$50x-150-4\times 3\left(2y+1\right)=5\left(6-4\left(x+y-1\right)\right)$$

Multiply -4 and 3 to get -12.

$$50x-150-12\left(2y+1\right)=5\left(6-4\left(x+y-1\right)\right)$$

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

$$50x-150-24y-12=5\left(6-4\left(x+y-1\right)\right)$$

Subtract 12 from -150 to get -162.

$$50x-162-24y=5\left(6-4\left(x+y-1\right)\right)$$

Use the distributive property to multiply -4 by x+y-1.

$$50x-162-24y=5\left(6-4x-4y+4\right)$$

Add 6 and 4 to get 10.

$$50x-162-24y=5\left(10-4x-4y\right)$$

Use the distributive property to multiply 5 by 10-4x-4y.

$$50x-162-24y=50-20x-20y$$

Add 20x to both sides.

$$50x-162-24y+20x=50-20y$$

Combine 50x and 20x to get 70x.

$$70x-162-24y=50-20y$$

Add 20y to both sides.

$$70x-162-24y+20y=50$$

Combine -24y and 20y to get -4y.

$$70x-162-4y=50$$

Add 162 to both sides.

$$70x-4y=50+162$$

Add 50 and 162 to get 212.

$$70x-4y=212$$

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

$$6x-10y+35=48$$

Subtract 35 from both sides.

$$6x-10y=48-35$$

Subtract 35 from 48 to get 13.

$$6x-10y=13$$

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

$$70x-4y=212,6x-10y=13$$

Write the equations in matrix form.

$$\left(\begin{matrix}70&-4\\6&-10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}212\\13\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}70&-4\\6&-10\end{matrix}\right).

$$inverse(\left(\begin{matrix}70&-4\\6&-10\end{matrix}\right))\left(\begin{matrix}70&-4\\6&-10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}70&-4\\6&-10\end{matrix}\right))\left(\begin{matrix}212\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}70&-4\\6&-10\end{matrix}\right))\left(\begin{matrix}212\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}70&-4\\6&-10\end{matrix}\right))\left(\begin{matrix}212\\13\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{10}{70\left(-10\right)-\left(-4\times 6\right)}&-\frac{-4}{70\left(-10\right)-\left(-4\times 6\right)}\\-\frac{6}{70\left(-10\right)-\left(-4\times 6\right)}&\frac{70}{70\left(-10\right)-\left(-4\times 6\right)}\end{matrix}\right)\left(\begin{matrix}212\\13\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{338}&-\frac{1}{169}\\\frac{3}{338}&-\frac{35}{338}\end{matrix}\right)\left(\begin{matrix}212\\13\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{338}\times 212-\frac{1}{169}\times 13\\\frac{3}{338}\times 212-\frac{35}{338}\times 13\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{517}{169}\\\frac{181}{338}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{517}{169},y=\frac{181}{338}$$

Steps Using Elimination

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

$$10\times 5\left(x-3\right)-4\times 3\left(2y+1\right)=5\left(6-4\left(x+y-1\right)\right)$$

Multiply 10 and 5 to get 50.

$$50\left(x-3\right)-4\times 3\left(2y+1\right)=5\left(6-4\left(x+y-1\right)\right)$$

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

$$50x-150-4\times 3\left(2y+1\right)=5\left(6-4\left(x+y-1\right)\right)$$

Multiply -4 and 3 to get -12.

$$50x-150-12\left(2y+1\right)=5\left(6-4\left(x+y-1\right)\right)$$

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

$$50x-150-24y-12=5\left(6-4\left(x+y-1\right)\right)$$

Subtract 12 from -150 to get -162.

$$50x-162-24y=5\left(6-4\left(x+y-1\right)\right)$$

Use the distributive property to multiply -4 by x+y-1.

$$50x-162-24y=5\left(6-4x-4y+4\right)$$

Add 6 and 4 to get 10.

$$50x-162-24y=5\left(10-4x-4y\right)$$

Use the distributive property to multiply 5 by 10-4x-4y.

$$50x-162-24y=50-20x-20y$$

Add 20x to both sides.

$$50x-162-24y+20x=50-20y$$

Combine 50x and 20x to get 70x.

$$70x-162-24y=50-20y$$

Add 20y to both sides.

$$70x-162-24y+20y=50$$

Combine -24y and 20y to get -4y.

$$70x-162-4y=50$$

Add 162 to both sides.

$$70x-4y=50+162$$

Add 50 and 162 to get 212.

$$70x-4y=212$$

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

$$6x-10y+35=48$$

Subtract 35 from both sides.

$$6x-10y=48-35$$

Subtract 35 from 48 to get 13.

$$6x-10y=13$$

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.

$$70x-4y=212,6x-10y=13$$

To make 70x 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 70.

$$6\times 70x+6\left(-4\right)y=6\times 212,70\times 6x+70\left(-10\right)y=70\times 13$$

Simplify.

$$420x-24y=1272,420x-700y=910$$

Subtract 420x-700y=910 from 420x-24y=1272 by subtracting like terms on each side of the equal sign.

$$420x-420x-24y+700y=1272-910$$

Add 420x to -420x. Terms 420x and -420x cancel out, leaving an equation with only one variable that can be solved.

$$-24y+700y=1272-910$$

Add -24y to 700y.

$$676y=1272-910$$

Add 1272 to -910.

$$676y=362$$

Divide both sides by 676.

$$y=\frac{181}{338}$$

Substitute \frac{181}{338} for y in 6x-10y=13. Because the resulting equation contains only one variable, you can solve for x directly.

$$6x-10\times \frac{181}{338}=13$$

Multiply -10 times \frac{181}{338}.

$$6x-\frac{905}{169}=13$$

Add \frac{905}{169} to both sides of the equation.

$$6x=\frac{3102}{169}$$

Divide both sides by 6.

$$x=\frac{517}{169}$$

The system is now solved.

$$x=\frac{517}{169},y=\frac{181}{338}$$