Solve {l}{3*[4x-3(x+1)]+8=2(y-x)}{5*[2x-4(x+1)]+24=6(x-y)}

$$\left\{ \begin{array} { l } { 3 \cdot [ 4 x - 3 ( x + 1 ) ] + 8 = 2 ( y - x ) } \\ { 5 \cdot [ 2 x - 4 ( x + 1 ) ] + 24 = 6 ( x - y ) } \end{array} \right.$$

Solve for x, y

$$x=1$$

$$y=2$$

Steps Using Substitution

Consider the first equation. Use the distributive property to multiply -3 by x+1.

$$3\left(4x-3x-3\right)+8=2\left(y-x\right)$$

Combine 4x and -3x to get x.

$$3\left(x-3\right)+8=2\left(y-x\right)$$

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

$$3x-9+8=2\left(y-x\right)$$

Add -9 and 8 to get -1.

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

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

$$3x-1=2y-2x$$

Subtract 2y from both sides.

$$3x-1-2y=-2x$$

Add 2x to both sides.

$$3x-1-2y+2x=0$$

Combine 3x and 2x to get 5x.

$$5x-1-2y=0$$

Add 1 to both sides. Anything plus zero gives itself.

$$5x-2y=1$$

Consider the second equation. Use the distributive property to multiply -4 by x+1.

$$5\left(2x-4x-4\right)+24=6\left(x-y\right)$$

Combine 2x and -4x to get -2x.

$$5\left(-2x-4\right)+24=6\left(x-y\right)$$

Use the distributive property to multiply 5 by -2x-4.

$$-10x-20+24=6\left(x-y\right)$$

Add -20 and 24 to get 4.

$$-10x+4=6\left(x-y\right)$$

Use the distributive property to multiply 6 by x-y.

$$-10x+4=6x-6y$$

Subtract 6x from both sides.

$$-10x+4-6x=-6y$$

Combine -10x and -6x to get -16x.

$$-16x+4=-6y$$

Add 6y to both sides.

$$-16x+4+6y=0$$

Subtract 4 from both sides. Anything subtracted from zero gives its negation.

$$-16x+6y=-4$$

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.

$$5x-2y=1,-16x+6y=-4$$

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

$$5x-2y=1$$

Add 2y to both sides of the equation.

$$5x=2y+1$$

Divide both sides by 5.

$$x=\frac{1}{5}\left(2y+1\right)$$

Multiply \frac{1}{5} times 2y+1.

$$x=\frac{2}{5}y+\frac{1}{5}$$

Substitute \frac{2y+1}{5} for x in the other equation, -16x+6y=-4.

$$-16\left(\frac{2}{5}y+\frac{1}{5}\right)+6y=-4$$

Multiply -16 times \frac{2y+1}{5}.

$$-\frac{32}{5}y-\frac{16}{5}+6y=-4$$

Add -\frac{32y}{5} to 6y.

$$-\frac{2}{5}y-\frac{16}{5}=-4$$

Add \frac{16}{5} to both sides of the equation.

$$-\frac{2}{5}y=-\frac{4}{5}$$

Divide both sides of the equation by -\frac{2}{5}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=2$$

Substitute 2 for y in x=\frac{2}{5}y+\frac{1}{5}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{2}{5}\times 2+\frac{1}{5}$$

Multiply \frac{2}{5} times 2.

$$x=\frac{4+1}{5}$$

Add \frac{1}{5} to \frac{4}{5} 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.

$$x=1,y=2$$

Steps Using Matrices

Consider the first equation. Use the distributive property to multiply -3 by x+1.

$$3\left(4x-3x-3\right)+8=2\left(y-x\right)$$

Combine 4x and -3x to get x.

$$3\left(x-3\right)+8=2\left(y-x\right)$$

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

$$3x-9+8=2\left(y-x\right)$$

Add -9 and 8 to get -1.

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

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

$$3x-1=2y-2x$$

Subtract 2y from both sides.

$$3x-1-2y=-2x$$

Add 2x to both sides.

$$3x-1-2y+2x=0$$

Combine 3x and 2x to get 5x.

$$5x-1-2y=0$$

Add 1 to both sides. Anything plus zero gives itself.

$$5x-2y=1$$

Consider the second equation. Use the distributive property to multiply -4 by x+1.

$$5\left(2x-4x-4\right)+24=6\left(x-y\right)$$

Combine 2x and -4x to get -2x.

$$5\left(-2x-4\right)+24=6\left(x-y\right)$$

Use the distributive property to multiply 5 by -2x-4.

$$-10x-20+24=6\left(x-y\right)$$

Add -20 and 24 to get 4.

$$-10x+4=6\left(x-y\right)$$

Use the distributive property to multiply 6 by x-y.

$$-10x+4=6x-6y$$

Subtract 6x from both sides.

$$-10x+4-6x=-6y$$

Combine -10x and -6x to get -16x.

$$-16x+4=-6y$$

Add 6y to both sides.

$$-16x+4+6y=0$$

Subtract 4 from both sides. Anything subtracted from zero gives its negation.

$$-16x+6y=-4$$

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

$$5x-2y=1,-16x+6y=-4$$

Write the equations in matrix form.

$$\left(\begin{matrix}5&-2\\-16&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\-4\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}5&-2\\-16&6\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&-2\\-16&6\end{matrix}\right))\left(\begin{matrix}5&-2\\-16&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-2\\-16&6\end{matrix}\right))\left(\begin{matrix}1\\-4\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-2\\-16&6\end{matrix}\right))\left(\begin{matrix}1\\-4\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-2\\-16&6\end{matrix}\right))\left(\begin{matrix}1\\-4\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{6}{5\times 6-\left(-2\left(-16\right)\right)}&-\frac{-2}{5\times 6-\left(-2\left(-16\right)\right)}\\-\frac{-16}{5\times 6-\left(-2\left(-16\right)\right)}&\frac{5}{5\times 6-\left(-2\left(-16\right)\right)}\end{matrix}\right)\left(\begin{matrix}1\\-4\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3&-1\\-8&-\frac{5}{2}\end{matrix}\right)\left(\begin{matrix}1\\-4\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3-\left(-4\right)\\-8-\frac{5}{2}\left(-4\right)\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\2\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=1,y=2$$

Steps Using Elimination

Consider the first equation. Use the distributive property to multiply -3 by x+1.

$$3\left(4x-3x-3\right)+8=2\left(y-x\right)$$

Combine 4x and -3x to get x.

$$3\left(x-3\right)+8=2\left(y-x\right)$$

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

$$3x-9+8=2\left(y-x\right)$$

Add -9 and 8 to get -1.

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

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

$$3x-1=2y-2x$$

Subtract 2y from both sides.

$$3x-1-2y=-2x$$

Add 2x to both sides.

$$3x-1-2y+2x=0$$

Combine 3x and 2x to get 5x.

$$5x-1-2y=0$$

Add 1 to both sides. Anything plus zero gives itself.

$$5x-2y=1$$

Consider the second equation. Use the distributive property to multiply -4 by x+1.

$$5\left(2x-4x-4\right)+24=6\left(x-y\right)$$

Combine 2x and -4x to get -2x.

$$5\left(-2x-4\right)+24=6\left(x-y\right)$$

Use the distributive property to multiply 5 by -2x-4.

$$-10x-20+24=6\left(x-y\right)$$

Add -20 and 24 to get 4.

$$-10x+4=6\left(x-y\right)$$

Use the distributive property to multiply 6 by x-y.

$$-10x+4=6x-6y$$

Subtract 6x from both sides.

$$-10x+4-6x=-6y$$

Combine -10x and -6x to get -16x.

$$-16x+4=-6y$$

Add 6y to both sides.

$$-16x+4+6y=0$$

Subtract 4 from both sides. Anything subtracted from zero gives its negation.

$$-16x+6y=-4$$

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.

$$5x-2y=1,-16x+6y=-4$$

To make 5x and -16x equal, multiply all terms on each side of the first equation by -16 and all terms on each side of the second by 5.

$$-16\times 5x-16\left(-2\right)y=-16,5\left(-16\right)x+5\times 6y=5\left(-4\right)$$

Simplify.

$$-80x+32y=-16,-80x+30y=-20$$

Subtract -80x+30y=-20 from -80x+32y=-16 by subtracting like terms on each side of the equal sign.

$$-80x+80x+32y-30y=-16+20$$

Add -80x to 80x. Terms -80x and 80x cancel out, leaving an equation with only one variable that can be solved.

$$32y-30y=-16+20$$

Add 32y to -30y.

$$2y=-16+20$$

Add -16 to 20.

$$2y=4$$

Divide both sides by 2.

$$y=2$$

Substitute 2 for y in -16x+6y=-4. Because the resulting equation contains only one variable, you can solve for x directly.

$$-16x+6\times 2=-4$$

Multiply 6 times 2.

$$-16x+12=-4$$

Subtract 12 from both sides of the equation.

$$-16x=-16$$

Divide both sides by -16.

$$x=1$$

The system is now solved.

$$x=1,y=2$$