Solve {l}{3(x+2)=2y}{2cy+5=7x}

$$\left\{ \begin{array} { l } { 3 ( x + 2 ) = 2 y } \\ { 2 c y + 5 = 7 x } \end{array} \right.$$

Solve for x, y

$$x=-\frac{6c+5}{3c-7}$$

$$y=-\frac{57}{2\left(3c-7\right)}$$

$$c\neq \frac{7}{3}$$

Steps Using Substitution

Consider the first equation. Use the distributive property to multiply 3 by x+2.

$$3x+6=2y$$

Subtract 2y from both sides.

$$3x+6-2y=0$$

Subtract 6 from both sides. Anything subtracted from zero gives its negation.

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

Consider the second equation. Subtract 7x from both sides.

$$2cy+5-7x=0$$

Subtract 5 from both sides. Anything subtracted from zero gives its negation.

$$2cy-7x=-5$$

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.

$$3x-2y=-6,-7x+2cy=-5$$

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

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

Add 2y to both sides of the equation.

$$3x=2y-6$$

Divide both sides by 3.

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

Multiply \frac{1}{3} times -6+2y.

$$x=\frac{2}{3}y-2$$

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

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

Multiply -7 times \frac{2y}{3}-2.

$$-\frac{14}{3}y+14+2cy=-5$$

Add -\frac{14y}{3} to 2cy.

$$\left(2c-\frac{14}{3}\right)y+14=-5$$

Subtract 14 from both sides of the equation.

$$\left(2c-\frac{14}{3}\right)y=-19$$

Divide both sides by -\frac{14}{3}+2c.

$$y=-\frac{57}{2\left(3c-7\right)}$$

Substitute -\frac{57}{2\left(-7+3c\right)} for y in x=\frac{2}{3}y-2. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{2}{3}\left(-\frac{57}{2\left(3c-7\right)}\right)-2$$

Multiply \frac{2}{3} times -\frac{57}{2\left(-7+3c\right)}.

$$x=-\frac{19}{3c-7}-2$$

Add -2 to -\frac{19}{-7+3c}.

$$x=-\frac{6c+5}{3c-7}$$

The system is now solved.

$$x=-\frac{6c+5}{3c-7},y=-\frac{57}{2\left(3c-7\right)}$$

Steps Using Matrices

Consider the first equation. Use the distributive property to multiply 3 by x+2.

$$3x+6=2y$$

Subtract 2y from both sides.

$$3x+6-2y=0$$

Subtract 6 from both sides. Anything subtracted from zero gives its negation.

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

Consider the second equation. Subtract 7x from both sides.

$$2cy+5-7x=0$$

Subtract 5 from both sides. Anything subtracted from zero gives its negation.

$$2cy-7x=-5$$

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

$$3x-2y=-6,-7x+2cy=-5$$

Write the equations in matrix form.

$$\left(\begin{matrix}3&-2\\-7&2c\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-6\\-5\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}3&-2\\-7&2c\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-2\\-7&2c\end{matrix}\right))\left(\begin{matrix}3&-2\\-7&2c\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\-7&2c\end{matrix}\right))\left(\begin{matrix}-6\\-5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\-7&2c\end{matrix}\right))\left(\begin{matrix}-6\\-5\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-2\\-7&2c\end{matrix}\right))\left(\begin{matrix}-6\\-5\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2c}{3\times 2c-\left(-2\left(-7\right)\right)}&-\frac{-2}{3\times 2c-\left(-2\left(-7\right)\right)}\\-\frac{-7}{3\times 2c-\left(-2\left(-7\right)\right)}&\frac{3}{3\times 2c-\left(-2\left(-7\right)\right)}\end{matrix}\right)\left(\begin{matrix}-6\\-5\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{c}{3c-7}&\frac{1}{3c-7}\\\frac{7}{2\left(3c-7\right)}&\frac{3}{2\left(3c-7\right)}\end{matrix}\right)\left(\begin{matrix}-6\\-5\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{c}{3c-7}\left(-6\right)+\frac{1}{3c-7}\left(-5\right)\\\frac{7}{2\left(3c-7\right)}\left(-6\right)+\frac{3}{2\left(3c-7\right)}\left(-5\right)\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{6c+5}{3c-7}\\-\frac{57}{2\left(3c-7\right)}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=-\frac{6c+5}{3c-7},y=-\frac{57}{2\left(3c-7\right)}$$

Steps Using Elimination

Consider the first equation. Use the distributive property to multiply 3 by x+2.

$$3x+6=2y$$

Subtract 2y from both sides.

$$3x+6-2y=0$$

Subtract 6 from both sides. Anything subtracted from zero gives its negation.

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

Consider the second equation. Subtract 7x from both sides.

$$2cy+5-7x=0$$

Subtract 5 from both sides. Anything subtracted from zero gives its negation.

$$2cy-7x=-5$$

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.

$$3x-2y=-6,-7x+2cy=-5$$

To make 3x and -7x equal, multiply all terms on each side of the first equation by -7 and all terms on each side of the second by 3.

$$-7\times 3x-7\left(-2\right)y=-7\left(-6\right),3\left(-7\right)x+3\times 2cy=3\left(-5\right)$$

Simplify.

$$-21x+14y=42,-21x+6cy=-15$$

Subtract -21x+6cy=-15 from -21x+14y=42 by subtracting like terms on each side of the equal sign.

$$-21x+21x+14y+\left(-6c\right)y=42+15$$

Add -21x to 21x. Terms -21x and 21x cancel out, leaving an equation with only one variable that can be solved.

$$14y+\left(-6c\right)y=42+15$$

Add 14y to -6cy.

$$\left(14-6c\right)y=42+15$$

Add 42 to 15.

$$\left(14-6c\right)y=57$$

Divide both sides by 14-6c.

$$y=\frac{57}{2\left(7-3c\right)}$$

Substitute \frac{57}{2\left(7-3c\right)} for y in -7x+2cy=-5. Because the resulting equation contains only one variable, you can solve for x directly.

$$-7x+2c\times \frac{57}{2\left(7-3c\right)}=-5$$

Multiply 2c times \frac{57}{2\left(7-3c\right)}.

$$-7x+\frac{57c}{7-3c}=-5$$

Subtract \frac{57c}{7-3c} from both sides of the equation.

$$-7x=-\frac{7\left(6c+5\right)}{7-3c}$$

Divide both sides by -7.

$$x=\frac{6c+5}{7-3c}$$

The system is now solved.

$$x=\frac{6c+5}{7-3c},y=\frac{57}{2\left(7-3c\right)}$$