Solve {l}{5(x+3y)-(7x+8y)=-6}{7x-9y-2(x-18y)=0}

$$\left. \begin{array} { l } { 5 ( x + 3 y ) - ( 7 x + 8 y ) = - 6 } \\ { 7 x - 9 y - 2 ( x - 18 y ) = 0 } \end{array} \right.$$

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 5 by x+3y.

$$5x+15y-\left(7x+8y\right)=-6$$

To find the opposite of 7x+8y, find the opposite of each term.

$$5x+15y-7x-8y=-6$$

Combine 5x and -7x to get -2x.

$$-2x+15y-8y=-6$$

Combine 15y and -8y to get 7y.

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

Consider the second equation. Use the distributive property to multiply -2 by x-18y.

$$7x-9y-2x+36y=0$$

Combine 7x and -2x to get 5x.

$$5x-9y+36y=0$$

Combine -9y and 36y to get 27y.

$$5x+27y=0$$

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.

$$-2x+7y=-6,5x+27y=0$$

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

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

Subtract 7y from both sides of the equation.

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

Divide both sides by -2.

$$x=-\frac{1}{2}\left(-7y-6\right)$$

Multiply -\frac{1}{2} times -7y-6.

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

Substitute \frac{7y}{2}+3 for x in the other equation, 5x+27y=0.

$$5\left(\frac{7}{2}y+3\right)+27y=0$$

Multiply 5 times \frac{7y}{2}+3.

$$\frac{35}{2}y+15+27y=0$$

Add \frac{35y}{2} to 27y.

$$\frac{89}{2}y+15=0$$

Subtract 15 from both sides of the equation.

$$\frac{89}{2}y=-15$$

Divide both sides of the equation by \frac{89}{2}, which is the same as multiplying both sides by the reciprocal of the fraction.

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

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 5 by x+3y.

$$5x+15y-\left(7x+8y\right)=-6$$

To find the opposite of 7x+8y, find the opposite of each term.

$$5x+15y-7x-8y=-6$$

Combine 5x and -7x to get -2x.

$$-2x+15y-8y=-6$$

Combine 15y and -8y to get 7y.

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

Consider the second equation. Use the distributive property to multiply -2 by x-18y.

$$7x-9y-2x+36y=0$$

Combine 7x and -2x to get 5x.

$$5x-9y+36y=0$$

Combine -9y and 36y to get 27y.

$$5x+27y=0$$

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

$$-2x+7y=-6,5x+27y=0$$

Write the equations in matrix form.

$$\left(\begin{matrix}-2&7\\5&27\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-6\\0\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}-2&7\\5&27\end{matrix}\right).

$$inverse(\left(\begin{matrix}-2&7\\5&27\end{matrix}\right))\left(\begin{matrix}-2&7\\5&27\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&7\\5&27\end{matrix}\right))\left(\begin{matrix}-6\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&7\\5&27\end{matrix}\right))\left(\begin{matrix}-6\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&7\\5&27\end{matrix}\right))\left(\begin{matrix}-6\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{27}{-2\times 27-7\times 5}&-\frac{7}{-2\times 27-7\times 5}\\-\frac{5}{-2\times 27-7\times 5}&-\frac{2}{-2\times 27-7\times 5}\end{matrix}\right)\left(\begin{matrix}-6\\0\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{27}{89}&\frac{7}{89}\\\frac{5}{89}&\frac{2}{89}\end{matrix}\right)\left(\begin{matrix}-6\\0\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{27}{89}\left(-6\right)\\\frac{5}{89}\left(-6\right)\end{matrix}\right)$$

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 5 by x+3y.

$$5x+15y-\left(7x+8y\right)=-6$$

To find the opposite of 7x+8y, find the opposite of each term.

$$5x+15y-7x-8y=-6$$

Combine 5x and -7x to get -2x.

$$-2x+15y-8y=-6$$

Combine 15y and -8y to get 7y.

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

Consider the second equation. Use the distributive property to multiply -2 by x-18y.

$$7x-9y-2x+36y=0$$

Combine 7x and -2x to get 5x.

$$5x-9y+36y=0$$

Combine -9y and 36y to get 27y.

$$5x+27y=0$$

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.

$$-2x+7y=-6,5x+27y=0$$

To make -2x 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 -2.

$$5\left(-2\right)x+5\times 7y=5\left(-6\right),-2\times 5x-2\times 27y=0$$

Simplify.

$$-10x+35y=-30,-10x-54y=0$$

Subtract -10x-54y=0 from -10x+35y=-30 by subtracting like terms on each side of the equal sign.

$$-10x+10x+35y+54y=-30$$

Add -10x to 10x. Terms -10x and 10x cancel out, leaving an equation with only one variable that can be solved.

$$35y+54y=-30$$

Add 35y to 54y.

$$89y=-30$$

Divide both sides by 89.

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

Substitute -\frac{30}{89} for y in 5x+27y=0. Because the resulting equation contains only one variable, you can solve for x directly.

$$5x+27\left(-\frac{30}{89}\right)=0$$

Multiply 27 times -\frac{30}{89}.

$$5x-\frac{810}{89}=0$$

Add \frac{810}{89} to both sides of the equation.

$$5x=\frac{810}{89}$$

Divide both sides by 5.

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

The system is now solved.

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