Solve {l}{3x-1/2-x-y/3=1}{x-y/4+x+2y/3=1}

$$\left\{ \begin{array} { l } { \frac { 3 x - 1 } { 2 } - \frac { x - y } { 3 } = 1 } \\ { \frac { x - y } { 4 } + \frac { x + 2 y } { 3 } = 1 } \end{array} \right.$$

Solve for x, y

$$x=1$$

$$y=1$$

Steps Using Substitution

Consider the first equation. Multiply both sides of the equation by 6, the least common multiple of 2,3.

$$3\left(3x-1\right)-2\left(x-y\right)=6$$

Use the distributive property to multiply 3 by 3x-1.

$$9x-3-2\left(x-y\right)=6$$

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

$$9x-3-2x+2y=6$$

Combine 9x and -2x to get 7x.

$$7x-3+2y=6$$

Add 3 to both sides.

$$7x+2y=6+3$$

Add 6 and 3 to get 9.

$$7x+2y=9$$

Consider the second equation. Multiply both sides of the equation by 12, the least common multiple of 4,3.

$$3\left(x-y\right)+4\left(x+2y\right)=12$$

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

$$3x-3y+4\left(x+2y\right)=12$$

Use the distributive property to multiply 4 by x+2y.

$$3x-3y+4x+8y=12$$

Combine 3x and 4x to get 7x.

$$7x-3y+8y=12$$

Combine -3y and 8y to get 5y.

$$7x+5y=12$$

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.

$$7x+2y=9,7x+5y=12$$

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

$$7x+2y=9$$

Subtract 2y from both sides of the equation.

$$7x=-2y+9$$

Divide both sides by 7.

$$x=\frac{1}{7}\left(-2y+9\right)$$

Multiply \frac{1}{7} times -2y+9.

$$x=-\frac{2}{7}y+\frac{9}{7}$$

Substitute \frac{-2y+9}{7} for x in the other equation, 7x+5y=12.

$$7\left(-\frac{2}{7}y+\frac{9}{7}\right)+5y=12$$

Multiply 7 times \frac{-2y+9}{7}.

$$-2y+9+5y=12$$

Add -2y to 5y.

$$3y+9=12$$

Subtract 9 from both sides of the equation.

$$3y=3$$

Divide both sides by 3.

$$y=1$$

Substitute 1 for y in x=-\frac{2}{7}y+\frac{9}{7}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{-2+9}{7}$$

Add \frac{9}{7} to -\frac{2}{7} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=1$$

The system is now solved.

$$x=1,y=1$$

Steps Using Matrices

Consider the first equation. Multiply both sides of the equation by 6, the least common multiple of 2,3.

$$3\left(3x-1\right)-2\left(x-y\right)=6$$

Use the distributive property to multiply 3 by 3x-1.

$$9x-3-2\left(x-y\right)=6$$

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

$$9x-3-2x+2y=6$$

Combine 9x and -2x to get 7x.

$$7x-3+2y=6$$

Add 3 to both sides.

$$7x+2y=6+3$$

Add 6 and 3 to get 9.

$$7x+2y=9$$

Consider the second equation. Multiply both sides of the equation by 12, the least common multiple of 4,3.

$$3\left(x-y\right)+4\left(x+2y\right)=12$$

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

$$3x-3y+4\left(x+2y\right)=12$$

Use the distributive property to multiply 4 by x+2y.

$$3x-3y+4x+8y=12$$

Combine 3x and 4x to get 7x.

$$7x-3y+8y=12$$

Combine -3y and 8y to get 5y.

$$7x+5y=12$$

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

$$7x+2y=9,7x+5y=12$$

Write the equations in matrix form.

$$\left(\begin{matrix}7&2\\7&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\12\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}7&2\\7&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}7&2\\7&5\end{matrix}\right))\left(\begin{matrix}7&2\\7&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&2\\7&5\end{matrix}\right))\left(\begin{matrix}9\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&2\\7&5\end{matrix}\right))\left(\begin{matrix}9\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&2\\7&5\end{matrix}\right))\left(\begin{matrix}9\\12\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{7\times 5-2\times 7}&-\frac{2}{7\times 5-2\times 7}\\-\frac{7}{7\times 5-2\times 7}&\frac{7}{7\times 5-2\times 7}\end{matrix}\right)\left(\begin{matrix}9\\12\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{21}&-\frac{2}{21}\\-\frac{1}{3}&\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}9\\12\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{21}\times 9-\frac{2}{21}\times 12\\-\frac{1}{3}\times 9+\frac{1}{3}\times 12\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\1\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=1,y=1$$

Steps Using Elimination

Consider the first equation. Multiply both sides of the equation by 6, the least common multiple of 2,3.

$$3\left(3x-1\right)-2\left(x-y\right)=6$$

Use the distributive property to multiply 3 by 3x-1.

$$9x-3-2\left(x-y\right)=6$$

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

$$9x-3-2x+2y=6$$

Combine 9x and -2x to get 7x.

$$7x-3+2y=6$$

Add 3 to both sides.

$$7x+2y=6+3$$

Add 6 and 3 to get 9.

$$7x+2y=9$$

Consider the second equation. Multiply both sides of the equation by 12, the least common multiple of 4,3.

$$3\left(x-y\right)+4\left(x+2y\right)=12$$

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

$$3x-3y+4\left(x+2y\right)=12$$

Use the distributive property to multiply 4 by x+2y.

$$3x-3y+4x+8y=12$$

Combine 3x and 4x to get 7x.

$$7x-3y+8y=12$$

Combine -3y and 8y to get 5y.

$$7x+5y=12$$

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.

$$7x+2y=9,7x+5y=12$$

Subtract 7x+5y=12 from 7x+2y=9 by subtracting like terms on each side of the equal sign.

$$7x-7x+2y-5y=9-12$$

Add 7x to -7x. Terms 7x and -7x cancel out, leaving an equation with only one variable that can be solved.

$$2y-5y=9-12$$

Add 2y to -5y.

$$-3y=9-12$$

Add 9 to -12.

$$-3y=-3$$

Divide both sides by -3.

$$y=1$$

Substitute 1 for y in 7x+5y=12. Because the resulting equation contains only one variable, you can solve for x directly.

$$7x+5=12$$

Subtract 5 from both sides of the equation.

$$7x=7$$

Divide both sides by 7.

$$x=1$$

The system is now solved.

$$x=1,y=1$$