Solve {l}{y=5/3-x/2}{2x/3+1/2=y}

$$\left. \begin{array} { l } { y = \frac { 5 } { 3 } - \frac { x } { 2 } } \\ { \frac { 2 x } { 3 } + \frac { 1 } { 2 } = y } \end{array} \right.$$

Solve for y, x

$$x=1$$

$$y = \frac{7}{6} = 1\frac{1}{6} \approx 1.166666667$$

Steps Using Substitution

Consider the first equation. To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 2 is 6. Multiply \frac{5}{3} times \frac{2}{2}. Multiply \frac{x}{2} times \frac{3}{3}.

$$y=\frac{5\times 2}{6}-\frac{3x}{6}$$

Since \frac{5\times 2}{6} and \frac{3x}{6} have the same denominator, subtract them by subtracting their numerators.

$$y=\frac{5\times 2-3x}{6}$$

Do the multiplications in 5\times 2-3x.

$$y=\frac{10-3x}{6}$$

Divide each term of 10-3x by 6 to get \frac{5}{3}-\frac{1}{2}x.

$$y=\frac{5}{3}-\frac{1}{2}x$$

Add \frac{1}{2}x to both sides.

$$y+\frac{1}{2}x=\frac{5}{3}$$

Consider the second equation. To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 2 is 6. Multiply \frac{2x}{3} times \frac{2}{2}. Multiply \frac{1}{2} times \frac{3}{3}.

$$\frac{2\times 2x}{6}+\frac{3}{6}=y$$

Since \frac{2\times 2x}{6} and \frac{3}{6} have the same denominator, add them by adding their numerators.

$$\frac{2\times 2x+3}{6}=y$$

Do the multiplications in 2\times 2x+3.

$$\frac{4x+3}{6}=y$$

Divide each term of 4x+3 by 6 to get \frac{2}{3}x+\frac{1}{2}.

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

Subtract y from both sides.

$$\frac{2}{3}x+\frac{1}{2}-y=0$$

Subtract \frac{1}{2} from both sides. Anything subtracted from zero gives its negation.

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

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.

$$y+\frac{1}{2}x=\frac{5}{3},-y+\frac{2}{3}x=-\frac{1}{2}$$

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

$$y+\frac{1}{2}x=\frac{5}{3}$$

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

$$y=-\frac{1}{2}x+\frac{5}{3}$$

Substitute -\frac{x}{2}+\frac{5}{3} for y in the other equation, -y+\frac{2}{3}x=-\frac{1}{2}.

$$-\left(-\frac{1}{2}x+\frac{5}{3}\right)+\frac{2}{3}x=-\frac{1}{2}$$

Multiply -1 times -\frac{x}{2}+\frac{5}{3}.

$$\frac{1}{2}x-\frac{5}{3}+\frac{2}{3}x=-\frac{1}{2}$$

Add \frac{x}{2} to \frac{2x}{3}.

$$\frac{7}{6}x-\frac{5}{3}=-\frac{1}{2}$$

Add \frac{5}{3} to both sides of the equation.

$$\frac{7}{6}x=\frac{7}{6}$$

Divide both sides of the equation by \frac{7}{6}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$x=1$$

Substitute 1 for x in y=-\frac{1}{2}x+\frac{5}{3}. Because the resulting equation contains only one variable, you can solve for y directly.

$$y=-\frac{1}{2}+\frac{5}{3}$$

Add \frac{5}{3} to -\frac{1}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y=\frac{7}{6}$$

The system is now solved.

$$y=\frac{7}{6},x=1$$

Steps Using Matrices

Consider the first equation. To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 2 is 6. Multiply \frac{5}{3} times \frac{2}{2}. Multiply \frac{x}{2} times \frac{3}{3}.

$$y=\frac{5\times 2}{6}-\frac{3x}{6}$$

Since \frac{5\times 2}{6} and \frac{3x}{6} have the same denominator, subtract them by subtracting their numerators.

$$y=\frac{5\times 2-3x}{6}$$

Do the multiplications in 5\times 2-3x.

$$y=\frac{10-3x}{6}$$

Divide each term of 10-3x by 6 to get \frac{5}{3}-\frac{1}{2}x.

$$y=\frac{5}{3}-\frac{1}{2}x$$

Add \frac{1}{2}x to both sides.

$$y+\frac{1}{2}x=\frac{5}{3}$$

Consider the second equation. To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 2 is 6. Multiply \frac{2x}{3} times \frac{2}{2}. Multiply \frac{1}{2} times \frac{3}{3}.

$$\frac{2\times 2x}{6}+\frac{3}{6}=y$$

Since \frac{2\times 2x}{6} and \frac{3}{6} have the same denominator, add them by adding their numerators.

$$\frac{2\times 2x+3}{6}=y$$

Do the multiplications in 2\times 2x+3.

$$\frac{4x+3}{6}=y$$

Divide each term of 4x+3 by 6 to get \frac{2}{3}x+\frac{1}{2}.

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

Subtract y from both sides.

$$\frac{2}{3}x+\frac{1}{2}-y=0$$

Subtract \frac{1}{2} from both sides. Anything subtracted from zero gives its negation.

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

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

$$y+\frac{1}{2}x=\frac{5}{3},-y+\frac{2}{3}x=-\frac{1}{2}$$

Write the equations in matrix form.

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

Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&\frac{1}{2}\\-1&\frac{2}{3}\end{matrix}\right).

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

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

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

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

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

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.

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

Do the arithmetic.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{4}{7}&-\frac{3}{7}\\\frac{6}{7}&\frac{6}{7}\end{matrix}\right)\left(\begin{matrix}\frac{5}{3}\\-\frac{1}{2}\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{4}{7}\times \frac{5}{3}-\frac{3}{7}\left(-\frac{1}{2}\right)\\\frac{6}{7}\times \frac{5}{3}+\frac{6}{7}\left(-\frac{1}{2}\right)\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{7}{6}\\1\end{matrix}\right)$$

Extract the matrix elements y and x.

$$y=\frac{7}{6},x=1$$

Steps Using Elimination

Consider the first equation. To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 2 is 6. Multiply \frac{5}{3} times \frac{2}{2}. Multiply \frac{x}{2} times \frac{3}{3}.

$$y=\frac{5\times 2}{6}-\frac{3x}{6}$$

Since \frac{5\times 2}{6} and \frac{3x}{6} have the same denominator, subtract them by subtracting their numerators.

$$y=\frac{5\times 2-3x}{6}$$

Do the multiplications in 5\times 2-3x.

$$y=\frac{10-3x}{6}$$

Divide each term of 10-3x by 6 to get \frac{5}{3}-\frac{1}{2}x.

$$y=\frac{5}{3}-\frac{1}{2}x$$

Add \frac{1}{2}x to both sides.

$$y+\frac{1}{2}x=\frac{5}{3}$$

Consider the second equation. To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 2 is 6. Multiply \frac{2x}{3} times \frac{2}{2}. Multiply \frac{1}{2} times \frac{3}{3}.

$$\frac{2\times 2x}{6}+\frac{3}{6}=y$$

Since \frac{2\times 2x}{6} and \frac{3}{6} have the same denominator, add them by adding their numerators.

$$\frac{2\times 2x+3}{6}=y$$

Do the multiplications in 2\times 2x+3.

$$\frac{4x+3}{6}=y$$

Divide each term of 4x+3 by 6 to get \frac{2}{3}x+\frac{1}{2}.

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

Subtract y from both sides.

$$\frac{2}{3}x+\frac{1}{2}-y=0$$

Subtract \frac{1}{2} from both sides. Anything subtracted from zero gives its negation.

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

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.

$$y+\frac{1}{2}x=\frac{5}{3},-y+\frac{2}{3}x=-\frac{1}{2}$$

To make y and -y equal, multiply all terms on each side of the first equation by -1 and all terms on each side of the second by 1.

$$-y-\frac{1}{2}x=-\frac{5}{3},-y+\frac{2}{3}x=-\frac{1}{2}$$

Subtract -y+\frac{2}{3}x=-\frac{1}{2} from -y-\frac{1}{2}x=-\frac{5}{3} by subtracting like terms on each side of the equal sign.

$$-y+y-\frac{1}{2}x-\frac{2}{3}x=-\frac{5}{3}+\frac{1}{2}$$

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

$$-\frac{1}{2}x-\frac{2}{3}x=-\frac{5}{3}+\frac{1}{2}$$

Add -\frac{x}{2} to -\frac{2x}{3}.

$$-\frac{7}{6}x=-\frac{5}{3}+\frac{1}{2}$$

Add -\frac{5}{3} to \frac{1}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$-\frac{7}{6}x=-\frac{7}{6}$$

Divide both sides of the equation by -\frac{7}{6}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$x=1$$

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

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

Subtract \frac{2}{3} from both sides of the equation.

$$-y=-\frac{7}{6}$$

Divide both sides by -1.

$$y=\frac{7}{6}$$

The system is now solved.

$$y=\frac{7}{6},x=1$$