Solve {l}{2x+3y=5}{7x=4y-3}

$$\left. \begin{array} { l } { 2 x + 3 y = 5 } \\ { 7 x = 4 y - 3 } \end{array} \right.$$

Solve for x, y

$$x=\frac{11}{29}\approx 0.379310345$$

$$y = \frac{41}{29} = 1\frac{12}{29} \approx 1.413793103$$

Steps Using Substitution

Consider the second equation. Subtract 4y from both sides.

$$7x-4y=-3$$

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.

$$2x+3y=5,7x-4y=-3$$

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

$$2x+3y=5$$

Subtract 3y from both sides of the equation.

$$2x=-3y+5$$

Divide both sides by 2.

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

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

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

Substitute \frac{-3y+5}{2} for x in the other equation, 7x-4y=-3.

$$7\left(-\frac{3}{2}y+\frac{5}{2}\right)-4y=-3$$

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

$$-\frac{21}{2}y+\frac{35}{2}-4y=-3$$

Add -\frac{21y}{2} to -4y.

$$-\frac{29}{2}y+\frac{35}{2}=-3$$

Subtract \frac{35}{2} from both sides of the equation.

$$-\frac{29}{2}y=-\frac{41}{2}$$

Divide both sides of the equation by -\frac{29}{2}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=\frac{41}{29}$$

Substitute \frac{41}{29} for y in x=-\frac{3}{2}y+\frac{5}{2}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{3}{2}\times \frac{41}{29}+\frac{5}{2}$$

Multiply -\frac{3}{2} times \frac{41}{29} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{123}{58}+\frac{5}{2}$$

Add \frac{5}{2} to -\frac{123}{58} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{11}{29}$$

The system is now solved.

$$x=\frac{11}{29},y=\frac{41}{29}$$

Steps Using Matrices

Consider the second equation. Subtract 4y from both sides.

$$7x-4y=-3$$

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

$$2x+3y=5,7x-4y=-3$$

Write the equations in matrix form.

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

Left multiply the equation by the inverse matrix of \left(\begin{matrix}2&3\\7&-4\end{matrix}\right).

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

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\7&-4\end{matrix}\right))\left(\begin{matrix}5\\-3\end{matrix}\right)$$

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

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

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.

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

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{29}&\frac{3}{29}\\\frac{7}{29}&-\frac{2}{29}\end{matrix}\right)\left(\begin{matrix}5\\-3\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{29}\times 5+\frac{3}{29}\left(-3\right)\\\frac{7}{29}\times 5-\frac{2}{29}\left(-3\right)\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{11}{29}\\\frac{41}{29}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{11}{29},y=\frac{41}{29}$$

Steps Using Elimination

Consider the second equation. Subtract 4y from both sides.

$$7x-4y=-3$$

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.

$$2x+3y=5,7x-4y=-3$$

To make 2x 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 2.

$$7\times 2x+7\times 3y=7\times 5,2\times 7x+2\left(-4\right)y=2\left(-3\right)$$

Simplify.

$$14x+21y=35,14x-8y=-6$$

Subtract 14x-8y=-6 from 14x+21y=35 by subtracting like terms on each side of the equal sign.

$$14x-14x+21y+8y=35+6$$

Add 14x to -14x. Terms 14x and -14x cancel out, leaving an equation with only one variable that can be solved.

$$21y+8y=35+6$$

Add 21y to 8y.

$$29y=35+6$$

Add 35 to 6.

$$29y=41$$

Divide both sides by 29.

$$y=\frac{41}{29}$$

Substitute \frac{41}{29} for y in 7x-4y=-3. Because the resulting equation contains only one variable, you can solve for x directly.

$$7x-4\times \frac{41}{29}=-3$$

Multiply -4 times \frac{41}{29}.

$$7x-\frac{164}{29}=-3$$

Add \frac{164}{29} to both sides of the equation.

$$7x=\frac{77}{29}$$

Divide both sides by 7.

$$x=\frac{11}{29}$$

The system is now solved.

$$x=\frac{11}{29},y=\frac{41}{29}$$