Solve {l}{4x+3y=10700}{3x+4y=10300}

$$\left. \begin{array} { l } { 4 x + 3 y = 10700 } \\ { 3 x + 4 y = 10300 } \end{array} \right.$$

Solve for x, y

$$x=1700$$

$$y=1300$$

Steps Using Substitution

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.

$$4x+3y=10700,3x+4y=10300$$

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

$$4x+3y=10700$$

Subtract 3y from both sides of the equation.

$$4x=-3y+10700$$

Divide both sides by 4.

$$x=\frac{1}{4}\left(-3y+10700\right)$$

Multiply \frac{1}{4} times -3y+10700.

$$x=-\frac{3}{4}y+2675$$

Substitute -\frac{3y}{4}+2675 for x in the other equation, 3x+4y=10300.

$$3\left(-\frac{3}{4}y+2675\right)+4y=10300$$

Multiply 3 times -\frac{3y}{4}+2675.

$$-\frac{9}{4}y+8025+4y=10300$$

Add -\frac{9y}{4} to 4y.

$$\frac{7}{4}y+8025=10300$$

Subtract 8025 from both sides of the equation.

$$\frac{7}{4}y=2275$$

Divide both sides of the equation by \frac{7}{4}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=1300$$

Substitute 1300 for y in x=-\frac{3}{4}y+2675. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{3}{4}\times 1300+2675$$

Multiply -\frac{3}{4} times 1300.

$$x=-975+2675$$

Add 2675 to -975.

$$x=1700$$

The system is now solved.

$$x=1700,y=1300$$

Steps Using Matrices

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

$$4x+3y=10700,3x+4y=10300$$

Write the equations in matrix form.

$$\left(\begin{matrix}4&3\\3&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10700\\10300\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}4&3\\3&4\end{matrix}\right))\left(\begin{matrix}4&3\\3&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\3&4\end{matrix}\right))\left(\begin{matrix}10700\\10300\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\3&4\end{matrix}\right))\left(\begin{matrix}10700\\10300\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\3&4\end{matrix}\right))\left(\begin{matrix}10700\\10300\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{4\times 4-3\times 3}&-\frac{3}{4\times 4-3\times 3}\\-\frac{3}{4\times 4-3\times 3}&\frac{4}{4\times 4-3\times 3}\end{matrix}\right)\left(\begin{matrix}10700\\10300\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{7}&-\frac{3}{7}\\-\frac{3}{7}&\frac{4}{7}\end{matrix}\right)\left(\begin{matrix}10700\\10300\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{7}\times 10700-\frac{3}{7}\times 10300\\-\frac{3}{7}\times 10700+\frac{4}{7}\times 10300\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1700\\1300\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=1700,y=1300$$

Steps Using Elimination

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.

$$4x+3y=10700,3x+4y=10300$$

To make 4x and 3x equal, multiply all terms on each side of the first equation by 3 and all terms on each side of the second by 4.

$$3\times 4x+3\times 3y=3\times 10700,4\times 3x+4\times 4y=4\times 10300$$

Simplify.

$$12x+9y=32100,12x+16y=41200$$

Subtract 12x+16y=41200 from 12x+9y=32100 by subtracting like terms on each side of the equal sign.

$$12x-12x+9y-16y=32100-41200$$

Add 12x to -12x. Terms 12x and -12x cancel out, leaving an equation with only one variable that can be solved.

$$9y-16y=32100-41200$$

Add 9y to -16y.

$$-7y=32100-41200$$

Add 32100 to -41200.

$$-7y=-9100$$

Divide both sides by -7.

$$y=1300$$

Substitute 1300 for y in 3x+4y=10300. Because the resulting equation contains only one variable, you can solve for x directly.

$$3x+4\times 1300=10300$$

Multiply 4 times 1300.

$$3x+5200=10300$$

Subtract 5200 from both sides of the equation.

$$3x=5100$$

Divide both sides by 3.

$$x=1700$$

The system is now solved.

$$x=1700,y=1300$$