Solve {l}{x-1/4-y+2/3=0}{x+3/5-y-2/4=2}

$$\left. \begin{array} { l } { \frac { x - 1 } { 4 } - \frac { y + 2 } { 3 } = 0 } \\ { \frac { x + 3 } { 5 } - \frac { y - 2 } { 4 } = 2 } \end{array} \right.$$

Solve for x, y

$$x=17$$

$$y=10$$

Steps Using Substitution

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

$$3\left(x-1\right)-4\left(y+2\right)=0$$

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

$$3x-3-4\left(y+2\right)=0$$

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

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

Subtract 8 from -3 to get -11.

$$3x-11-4y=0$$

Add 11 to both sides. Anything plus zero gives itself.

$$3x-4y=11$$

Consider the second equation. Multiply both sides of the equation by 20, the least common multiple of 5,4.

$$4\left(x+3\right)-5\left(y-2\right)=40$$

Use the distributive property to multiply 4 by x+3.

$$4x+12-5\left(y-2\right)=40$$

Use the distributive property to multiply -5 by y-2.

$$4x+12-5y+10=40$$

Add 12 and 10 to get 22.

$$4x+22-5y=40$$

Subtract 22 from both sides.

$$4x-5y=40-22$$

Subtract 22 from 40 to get 18.

$$4x-5y=18$$

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.

$$3x-4y=11,4x-5y=18$$

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

$$3x-4y=11$$

Add 4y to both sides of the equation.

$$3x=4y+11$$

Divide both sides by 3.

$$x=\frac{1}{3}\left(4y+11\right)$$

Multiply \frac{1}{3} times 4y+11.

$$x=\frac{4}{3}y+\frac{11}{3}$$

Substitute \frac{4y+11}{3} for x in the other equation, 4x-5y=18.

$$4\left(\frac{4}{3}y+\frac{11}{3}\right)-5y=18$$

Multiply 4 times \frac{4y+11}{3}.

$$\frac{16}{3}y+\frac{44}{3}-5y=18$$

Add \frac{16y}{3} to -5y.

$$\frac{1}{3}y+\frac{44}{3}=18$$

Subtract \frac{44}{3} from both sides of the equation.

$$\frac{1}{3}y=\frac{10}{3}$$

Multiply both sides by 3.

$$y=10$$

Substitute 10 for y in x=\frac{4}{3}y+\frac{11}{3}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{4}{3}\times 10+\frac{11}{3}$$

Multiply \frac{4}{3} times 10.

$$x=\frac{40+11}{3}$$

Add \frac{11}{3} to \frac{40}{3} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=17$$

The system is now solved.

$$x=17,y=10$$

Steps Using Matrices

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

$$3\left(x-1\right)-4\left(y+2\right)=0$$

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

$$3x-3-4\left(y+2\right)=0$$

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

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

Subtract 8 from -3 to get -11.

$$3x-11-4y=0$$

Add 11 to both sides. Anything plus zero gives itself.

$$3x-4y=11$$

Consider the second equation. Multiply both sides of the equation by 20, the least common multiple of 5,4.

$$4\left(x+3\right)-5\left(y-2\right)=40$$

Use the distributive property to multiply 4 by x+3.

$$4x+12-5\left(y-2\right)=40$$

Use the distributive property to multiply -5 by y-2.

$$4x+12-5y+10=40$$

Add 12 and 10 to get 22.

$$4x+22-5y=40$$

Subtract 22 from both sides.

$$4x-5y=40-22$$

Subtract 22 from 40 to get 18.

$$4x-5y=18$$

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

$$3x-4y=11,4x-5y=18$$

Write the equations in matrix form.

$$\left(\begin{matrix}3&-4\\4&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}11\\18\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}3&-4\\4&-5\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-4\\4&-5\end{matrix}\right))\left(\begin{matrix}3&-4\\4&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-4\\4&-5\end{matrix}\right))\left(\begin{matrix}11\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-4\\4&-5\end{matrix}\right))\left(\begin{matrix}11\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-4\\4&-5\end{matrix}\right))\left(\begin{matrix}11\\18\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{3\left(-5\right)-\left(-4\times 4\right)}&-\frac{-4}{3\left(-5\right)-\left(-4\times 4\right)}\\-\frac{4}{3\left(-5\right)-\left(-4\times 4\right)}&\frac{3}{3\left(-5\right)-\left(-4\times 4\right)}\end{matrix}\right)\left(\begin{matrix}11\\18\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-5&4\\-4&3\end{matrix}\right)\left(\begin{matrix}11\\18\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-5\times 11+4\times 18\\-4\times 11+3\times 18\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}17\\10\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=17,y=10$$

Steps Using Elimination

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

$$3\left(x-1\right)-4\left(y+2\right)=0$$

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

$$3x-3-4\left(y+2\right)=0$$

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

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

Subtract 8 from -3 to get -11.

$$3x-11-4y=0$$

Add 11 to both sides. Anything plus zero gives itself.

$$3x-4y=11$$

Consider the second equation. Multiply both sides of the equation by 20, the least common multiple of 5,4.

$$4\left(x+3\right)-5\left(y-2\right)=40$$

Use the distributive property to multiply 4 by x+3.

$$4x+12-5\left(y-2\right)=40$$

Use the distributive property to multiply -5 by y-2.

$$4x+12-5y+10=40$$

Add 12 and 10 to get 22.

$$4x+22-5y=40$$

Subtract 22 from both sides.

$$4x-5y=40-22$$

Subtract 22 from 40 to get 18.

$$4x-5y=18$$

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.

$$3x-4y=11,4x-5y=18$$

To make 3x and 4x equal, multiply all terms on each side of the first equation by 4 and all terms on each side of the second by 3.

$$4\times 3x+4\left(-4\right)y=4\times 11,3\times 4x+3\left(-5\right)y=3\times 18$$

Simplify.

$$12x-16y=44,12x-15y=54$$

Subtract 12x-15y=54 from 12x-16y=44 by subtracting like terms on each side of the equal sign.

$$12x-12x-16y+15y=44-54$$

Add 12x to -12x. Terms 12x and -12x cancel out, leaving an equation with only one variable that can be solved.

$$-16y+15y=44-54$$

Add -16y to 15y.

$$-y=44-54$$

Add 44 to -54.

$$-y=-10$$

Divide both sides by -1.

$$y=10$$

Substitute 10 for y in 4x-5y=18. Because the resulting equation contains only one variable, you can solve for x directly.

$$4x-5\times 10=18$$

Multiply -5 times 10.

$$4x-50=18$$

Add 50 to both sides of the equation.

$$4x=68$$

Divide both sides by 4.

$$x=17$$

The system is now solved.

$$x=17,y=10$$