Solve {c}{10(x+3y)-2(7x+8y)=-12}{14x-18y-4(x-18y)=0}

$$\left\{ \begin{array} { c } { 10 ( x + 3 y ) - 2 ( 7 x + 8 y ) = - 12 } \\ { 14 x - 18 y - 4 ( x - 18 y ) = 0 } \end{array} \right.$$

Solve for x, y

$$x = \frac{162}{89} = 1\frac{73}{89} \approx 1.820224719$$

$$y=-\frac{30}{89}\approx -0.337078652$$

Steps Using Substitution

Consider the first equation. Use the distributive property to multiply 10 by x+3y.

$$10x+30y-2\left(7x+8y\right)=-12$$

Use the distributive property to multiply -2 by 7x+8y.

$$10x+30y-14x-16y=-12$$

Combine 10x and -14x to get -4x.

$$-4x+30y-16y=-12$$

Combine 30y and -16y to get 14y.

$$-4x+14y=-12$$

Consider the second equation. Use the distributive property to multiply -4 by x-18y.

$$14x-18y-4x+72y=0$$

Combine 14x and -4x to get 10x.

$$10x-18y+72y=0$$

Combine -18y and 72y to get 54y.

$$10x+54y=0$$

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.

$$-4x+14y=-12,10x+54y=0$$

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

$$-4x+14y=-12$$

Subtract 14y from both sides of the equation.

$$-4x=-14y-12$$

Divide both sides by -4.

$$x=-\frac{1}{4}\left(-14y-12\right)$$

Multiply -\frac{1}{4} times -14y-12.

$$x=\frac{7}{2}y+3$$

Substitute \frac{7y}{2}+3 for x in the other equation, 10x+54y=0.

$$10\left(\frac{7}{2}y+3\right)+54y=0$$

Multiply 10 times \frac{7y}{2}+3.

$$35y+30+54y=0$$

Add 35y to 54y.

$$89y+30=0$$

Subtract 30 from both sides of the equation.

$$89y=-30$$

Divide both sides by 89.

$$y=-\frac{30}{89}$$

Substitute -\frac{30}{89} for y in x=\frac{7}{2}y+3. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{7}{2}\left(-\frac{30}{89}\right)+3$$

Multiply \frac{7}{2} times -\frac{30}{89} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{105}{89}+3$$

Add 3 to -\frac{105}{89}.

$$x=\frac{162}{89}$$

The system is now solved.

$$x=\frac{162}{89},y=-\frac{30}{89}$$

Steps Using Matrices

Consider the first equation. Use the distributive property to multiply 10 by x+3y.

$$10x+30y-2\left(7x+8y\right)=-12$$

Use the distributive property to multiply -2 by 7x+8y.

$$10x+30y-14x-16y=-12$$

Combine 10x and -14x to get -4x.

$$-4x+30y-16y=-12$$

Combine 30y and -16y to get 14y.

$$-4x+14y=-12$$

Consider the second equation. Use the distributive property to multiply -4 by x-18y.

$$14x-18y-4x+72y=0$$

Combine 14x and -4x to get 10x.

$$10x-18y+72y=0$$

Combine -18y and 72y to get 54y.

$$10x+54y=0$$

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

$$-4x+14y=-12,10x+54y=0$$

Write the equations in matrix form.

$$\left(\begin{matrix}-4&14\\10&54\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-12\\0\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}-4&14\\10&54\end{matrix}\right).

$$inverse(\left(\begin{matrix}-4&14\\10&54\end{matrix}\right))\left(\begin{matrix}-4&14\\10&54\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-4&14\\10&54\end{matrix}\right))\left(\begin{matrix}-12\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-4&14\\10&54\end{matrix}\right))\left(\begin{matrix}-12\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-4&14\\10&54\end{matrix}\right))\left(\begin{matrix}-12\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{54}{-4\times 54-14\times 10}&-\frac{14}{-4\times 54-14\times 10}\\-\frac{10}{-4\times 54-14\times 10}&-\frac{4}{-4\times 54-14\times 10}\end{matrix}\right)\left(\begin{matrix}-12\\0\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{27}{178}&\frac{7}{178}\\\frac{5}{178}&\frac{1}{89}\end{matrix}\right)\left(\begin{matrix}-12\\0\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{27}{178}\left(-12\right)\\\frac{5}{178}\left(-12\right)\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{162}{89}\\-\frac{30}{89}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{162}{89},y=-\frac{30}{89}$$

Steps Using Elimination

Consider the first equation. Use the distributive property to multiply 10 by x+3y.

$$10x+30y-2\left(7x+8y\right)=-12$$

Use the distributive property to multiply -2 by 7x+8y.

$$10x+30y-14x-16y=-12$$

Combine 10x and -14x to get -4x.

$$-4x+30y-16y=-12$$

Combine 30y and -16y to get 14y.

$$-4x+14y=-12$$

Consider the second equation. Use the distributive property to multiply -4 by x-18y.

$$14x-18y-4x+72y=0$$

Combine 14x and -4x to get 10x.

$$10x-18y+72y=0$$

Combine -18y and 72y to get 54y.

$$10x+54y=0$$

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.

$$-4x+14y=-12,10x+54y=0$$

To make -4x and 10x equal, multiply all terms on each side of the first equation by 10 and all terms on each side of the second by -4.

$$10\left(-4\right)x+10\times 14y=10\left(-12\right),-4\times 10x-4\times 54y=0$$

Simplify.

$$-40x+140y=-120,-40x-216y=0$$

Subtract -40x-216y=0 from -40x+140y=-120 by subtracting like terms on each side of the equal sign.

$$-40x+40x+140y+216y=-120$$

Add -40x to 40x. Terms -40x and 40x cancel out, leaving an equation with only one variable that can be solved.

$$140y+216y=-120$$

Add 140y to 216y.

$$356y=-120$$

Divide both sides by 356.

$$y=-\frac{30}{89}$$

Substitute -\frac{30}{89} for y in 10x+54y=0. Because the resulting equation contains only one variable, you can solve for x directly.

$$10x+54\left(-\frac{30}{89}\right)=0$$

Multiply 54 times -\frac{30}{89}.

$$10x-\frac{1620}{89}=0$$

Add \frac{1620}{89} to both sides of the equation.

$$10x=\frac{1620}{89}$$

Divide both sides by 10.

$$x=\frac{162}{89}$$

The system is now solved.

$$x=\frac{162}{89},y=-\frac{30}{89}$$