Solve {l}{5x+1/2-9y+6/4=2}{5(x-1)-4(y-2)=18}

$$\left. \begin{array} { l } { \frac { 5 x + 1 } { 2 } - \frac { 9 y + 6 } { 4 } = 2 } \\ { 5 ( x - 1 ) - 4 ( y - 2 ) = 18 } \end{array} \right.$$

Solve for x, y

$$x = \frac{87}{5} = 17\frac{2}{5} = 17.4$$

$$y=18$$

Steps Using Substitution

Consider the first equation. Multiply both sides of the equation by 4, the least common multiple of 2,4.

$$2\left(5x+1\right)-\left(9y+6\right)=8$$

Use the distributive property to multiply 2 by 5x+1.

$$10x+2-\left(9y+6\right)=8$$

To find the opposite of 9y+6, find the opposite of each term.

$$10x+2-9y-6=8$$

Subtract 6 from 2 to get -4.

$$10x-4-9y=8$$

Add 4 to both sides.

$$10x-9y=8+4$$

Add 8 and 4 to get 12.

$$10x-9y=12$$

Consider the second equation. Use the distributive property to multiply 5 by x-1.

$$5x-5-4\left(y-2\right)=18$$

Use the distributive property to multiply -4 by y-2.

$$5x-5-4y+8=18$$

Add -5 and 8 to get 3.

$$5x+3-4y=18$$

Subtract 3 from both sides.

$$5x-4y=18-3$$

Subtract 3 from 18 to get 15.

$$5x-4y=15$$

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.

$$10x-9y=12,5x-4y=15$$

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

$$10x-9y=12$$

Add 9y to both sides of the equation.

$$10x=9y+12$$

Divide both sides by 10.

$$x=\frac{1}{10}\left(9y+12\right)$$

Multiply \frac{1}{10} times 9y+12.

$$x=\frac{9}{10}y+\frac{6}{5}$$

Substitute \frac{9y}{10}+\frac{6}{5} for x in the other equation, 5x-4y=15.

$$5\left(\frac{9}{10}y+\frac{6}{5}\right)-4y=15$$

Multiply 5 times \frac{9y}{10}+\frac{6}{5}.

$$\frac{9}{2}y+6-4y=15$$

Add \frac{9y}{2} to -4y.

$$\frac{1}{2}y+6=15$$

Subtract 6 from both sides of the equation.

$$\frac{1}{2}y=9$$

Multiply both sides by 2.

$$y=18$$

Substitute 18 for y in x=\frac{9}{10}y+\frac{6}{5}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{9}{10}\times 18+\frac{6}{5}$$

Multiply \frac{9}{10} times 18.

$$x=\frac{81+6}{5}$$

Add \frac{6}{5} to \frac{81}{5} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{87}{5}$$

The system is now solved.

$$x=\frac{87}{5},y=18$$

Steps Using Matrices

Consider the first equation. Multiply both sides of the equation by 4, the least common multiple of 2,4.

$$2\left(5x+1\right)-\left(9y+6\right)=8$$

Use the distributive property to multiply 2 by 5x+1.

$$10x+2-\left(9y+6\right)=8$$

To find the opposite of 9y+6, find the opposite of each term.

$$10x+2-9y-6=8$$

Subtract 6 from 2 to get -4.

$$10x-4-9y=8$$

Add 4 to both sides.

$$10x-9y=8+4$$

Add 8 and 4 to get 12.

$$10x-9y=12$$

Consider the second equation. Use the distributive property to multiply 5 by x-1.

$$5x-5-4\left(y-2\right)=18$$

Use the distributive property to multiply -4 by y-2.

$$5x-5-4y+8=18$$

Add -5 and 8 to get 3.

$$5x+3-4y=18$$

Subtract 3 from both sides.

$$5x-4y=18-3$$

Subtract 3 from 18 to get 15.

$$5x-4y=15$$

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

$$10x-9y=12,5x-4y=15$$

Write the equations in matrix form.

$$\left(\begin{matrix}10&-9\\5&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}12\\15\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}10&-9\\5&-4\end{matrix}\right).

$$inverse(\left(\begin{matrix}10&-9\\5&-4\end{matrix}\right))\left(\begin{matrix}10&-9\\5&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&-9\\5&-4\end{matrix}\right))\left(\begin{matrix}12\\15\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&-9\\5&-4\end{matrix}\right))\left(\begin{matrix}12\\15\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&-9\\5&-4\end{matrix}\right))\left(\begin{matrix}12\\15\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{10\left(-4\right)-\left(-9\times 5\right)}&-\frac{-9}{10\left(-4\right)-\left(-9\times 5\right)}\\-\frac{5}{10\left(-4\right)-\left(-9\times 5\right)}&\frac{10}{10\left(-4\right)-\left(-9\times 5\right)}\end{matrix}\right)\left(\begin{matrix}12\\15\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{5}&\frac{9}{5}\\-1&2\end{matrix}\right)\left(\begin{matrix}12\\15\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{5}\times 12+\frac{9}{5}\times 15\\-12+2\times 15\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{87}{5}\\18\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{87}{5},y=18$$

Steps Using Elimination

Consider the first equation. Multiply both sides of the equation by 4, the least common multiple of 2,4.

$$2\left(5x+1\right)-\left(9y+6\right)=8$$

Use the distributive property to multiply 2 by 5x+1.

$$10x+2-\left(9y+6\right)=8$$

To find the opposite of 9y+6, find the opposite of each term.

$$10x+2-9y-6=8$$

Subtract 6 from 2 to get -4.

$$10x-4-9y=8$$

Add 4 to both sides.

$$10x-9y=8+4$$

Add 8 and 4 to get 12.

$$10x-9y=12$$

Consider the second equation. Use the distributive property to multiply 5 by x-1.

$$5x-5-4\left(y-2\right)=18$$

Use the distributive property to multiply -4 by y-2.

$$5x-5-4y+8=18$$

Add -5 and 8 to get 3.

$$5x+3-4y=18$$

Subtract 3 from both sides.

$$5x-4y=18-3$$

Subtract 3 from 18 to get 15.

$$5x-4y=15$$

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.

$$10x-9y=12,5x-4y=15$$

To make 10x and 5x equal, multiply all terms on each side of the first equation by 5 and all terms on each side of the second by 10.

$$5\times 10x+5\left(-9\right)y=5\times 12,10\times 5x+10\left(-4\right)y=10\times 15$$

Simplify.

$$50x-45y=60,50x-40y=150$$

Subtract 50x-40y=150 from 50x-45y=60 by subtracting like terms on each side of the equal sign.

$$50x-50x-45y+40y=60-150$$

Add 50x to -50x. Terms 50x and -50x cancel out, leaving an equation with only one variable that can be solved.

$$-45y+40y=60-150$$

Add -45y to 40y.

$$-5y=60-150$$

Add 60 to -150.

$$-5y=-90$$

Divide both sides by -5.

$$y=18$$

Substitute 18 for y in 5x-4y=15. Because the resulting equation contains only one variable, you can solve for x directly.

$$5x-4\times 18=15$$

Multiply -4 times 18.

$$5x-72=15$$

Add 72 to both sides of the equation.

$$5x=87$$

Divide both sides by 5.

$$x=\frac{87}{5}$$

The system is now solved.

$$x=\frac{87}{5},y=18$$