Solve {l}{y=x+3/2+3}{y=2x+10}

$$\left. \begin{array} { l } { y = \frac { x + 3 } { 2 } + 3 } \\ { y = 2 x + 10 } \end{array} \right.$$

Solve for y, x

$$x = -\frac{11}{3} = -3\frac{2}{3} \approx -3.666666667$$

$$y = \frac{8}{3} = 2\frac{2}{3} \approx 2.666666667$$

Steps Using Substitution

Consider the first equation. Divide each term of x+3 by 2 to get \frac{1}{2}x+\frac{3}{2}.

$$y=\frac{1}{2}x+\frac{3}{2}+3$$

Add \frac{3}{2} and 3 to get \frac{9}{2}.

$$y=\frac{1}{2}x+\frac{9}{2}$$

Substitute \frac{9+x}{2} for y in the other equation, y-2x=10.

$$\frac{1}{2}x+\frac{9}{2}-2x=10$$

Add \frac{x}{2} to -2x.

$$-\frac{3}{2}x+\frac{9}{2}=10$$

Subtract \frac{9}{2} from both sides of the equation.

$$-\frac{3}{2}x=\frac{11}{2}$$

Divide both sides of the equation by -\frac{3}{2}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$x=-\frac{11}{3}$$

Substitute -\frac{11}{3} for x in y=\frac{1}{2}x+\frac{9}{2}. Because the resulting equation contains only one variable, you can solve for y directly.

$$y=\frac{1}{2}\left(-\frac{11}{3}\right)+\frac{9}{2}$$

Multiply \frac{1}{2} times -\frac{11}{3} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$y=-\frac{11}{6}+\frac{9}{2}$$

Add \frac{9}{2} to -\frac{11}{6} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y=\frac{8}{3}$$

The system is now solved.

$$y=\frac{8}{3},x=-\frac{11}{3}$$

Steps Using Matrices

Consider the first equation. Divide each term of x+3 by 2 to get \frac{1}{2}x+\frac{3}{2}.

$$y=\frac{1}{2}x+\frac{3}{2}+3$$

Add \frac{3}{2} and 3 to get \frac{9}{2}.

$$y=\frac{1}{2}x+\frac{9}{2}$$

Subtract \frac{1}{2}x from both sides.

$$y-\frac{1}{2}x=\frac{9}{2}$$

Consider the second equation. Subtract 2x from both sides.

$$y-2x=10$$

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

$$y-\frac{1}{2}x=\frac{9}{2},y-2x=10$$

Write the equations in matrix form.

$$\left(\begin{matrix}1&-\frac{1}{2}\\1&-2\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{9}{2}\\10\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&-\frac{1}{2}\\1&-2\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-\frac{1}{2}\\1&-2\end{matrix}\right))\left(\begin{matrix}1&-\frac{1}{2}\\1&-2\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-\frac{1}{2}\\1&-2\end{matrix}\right))\left(\begin{matrix}\frac{9}{2}\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-\frac{1}{2}\\1&-2\end{matrix}\right))\left(\begin{matrix}\frac{9}{2}\\10\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-\frac{1}{2}\\1&-2\end{matrix}\right))\left(\begin{matrix}\frac{9}{2}\\10\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{-2-\left(-\frac{1}{2}\right)}&-\frac{-\frac{1}{2}}{-2-\left(-\frac{1}{2}\right)}\\-\frac{1}{-2-\left(-\frac{1}{2}\right)}&\frac{1}{-2-\left(-\frac{1}{2}\right)}\end{matrix}\right)\left(\begin{matrix}\frac{9}{2}\\10\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{4}{3}&-\frac{1}{3}\\\frac{2}{3}&-\frac{2}{3}\end{matrix}\right)\left(\begin{matrix}\frac{9}{2}\\10\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{4}{3}\times \frac{9}{2}-\frac{1}{3}\times 10\\\frac{2}{3}\times \frac{9}{2}-\frac{2}{3}\times 10\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{8}{3}\\-\frac{11}{3}\end{matrix}\right)$$

Extract the matrix elements y and x.

$$y=\frac{8}{3},x=-\frac{11}{3}$$

Steps Using Elimination

Consider the first equation. Divide each term of x+3 by 2 to get \frac{1}{2}x+\frac{3}{2}.

$$y=\frac{1}{2}x+\frac{3}{2}+3$$

Add \frac{3}{2} and 3 to get \frac{9}{2}.

$$y=\frac{1}{2}x+\frac{9}{2}$$

Subtract \frac{1}{2}x from both sides.

$$y-\frac{1}{2}x=\frac{9}{2}$$

Consider the second equation. Subtract 2x from both sides.

$$y-2x=10$$

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.

$$y-\frac{1}{2}x=\frac{9}{2},y-2x=10$$

Subtract y-2x=10 from y-\frac{1}{2}x=\frac{9}{2} by subtracting like terms on each side of the equal sign.

$$y-y-\frac{1}{2}x+2x=\frac{9}{2}-10$$

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

$$-\frac{1}{2}x+2x=\frac{9}{2}-10$$

Add -\frac{x}{2} to 2x.

$$\frac{3}{2}x=\frac{9}{2}-10$$

Add \frac{9}{2} to -10.

$$\frac{3}{2}x=-\frac{11}{2}$$

Divide both sides of the equation by \frac{3}{2}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$x=-\frac{11}{3}$$

Substitute -\frac{11}{3} for x in y-2x=10. Because the resulting equation contains only one variable, you can solve for y directly.

$$y-2\left(-\frac{11}{3}\right)=10$$

Multiply -2 times -\frac{11}{3}.

$$y+\frac{22}{3}=10$$

Subtract \frac{22}{3} from both sides of the equation.

$$y=\frac{8}{3}$$

The system is now solved.

$$y=\frac{8}{3},x=-\frac{11}{3}$$