Solve {l}{7x+16y=414}{10x+12y=548}

$$\left\{ \begin{array} { l } { 7 x + 16 y = 414 } \\ { 10 x + 12 y = 548 } \end{array} \right.$$

Solve for x, y

$$x=50$$

$$y=4$$

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.

$$7x+16y=414,10x+12y=548$$

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

$$7x+16y=414$$

Subtract 16y from both sides of the equation.

$$7x=-16y+414$$

Divide both sides by 7.

$$x=\frac{1}{7}\left(-16y+414\right)$$

Multiply \frac{1}{7} times -16y+414.

$$x=-\frac{16}{7}y+\frac{414}{7}$$

Substitute \frac{-16y+414}{7} for x in the other equation, 10x+12y=548.

$$10\left(-\frac{16}{7}y+\frac{414}{7}\right)+12y=548$$

Multiply 10 times \frac{-16y+414}{7}.

$$-\frac{160}{7}y+\frac{4140}{7}+12y=548$$

Add -\frac{160y}{7} to 12y.

$$-\frac{76}{7}y+\frac{4140}{7}=548$$

Subtract \frac{4140}{7} from both sides of the equation.

$$-\frac{76}{7}y=-\frac{304}{7}$$

Divide both sides of the equation by -\frac{76}{7}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=4$$

Substitute 4 for y in x=-\frac{16}{7}y+\frac{414}{7}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{16}{7}\times 4+\frac{414}{7}$$

Multiply -\frac{16}{7} times 4.

$$x=\frac{-64+414}{7}$$

Add \frac{414}{7} to -\frac{64}{7} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=50$$

The system is now solved.

$$x=50,y=4$$

Steps Using Matrices

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

$$7x+16y=414,10x+12y=548$$

Write the equations in matrix form.

$$\left(\begin{matrix}7&16\\10&12\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}414\\548\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}7&16\\10&12\end{matrix}\right).

$$inverse(\left(\begin{matrix}7&16\\10&12\end{matrix}\right))\left(\begin{matrix}7&16\\10&12\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&16\\10&12\end{matrix}\right))\left(\begin{matrix}414\\548\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&16\\10&12\end{matrix}\right))\left(\begin{matrix}414\\548\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&16\\10&12\end{matrix}\right))\left(\begin{matrix}414\\548\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{12}{7\times 12-16\times 10}&-\frac{16}{7\times 12-16\times 10}\\-\frac{10}{7\times 12-16\times 10}&\frac{7}{7\times 12-16\times 10}\end{matrix}\right)\left(\begin{matrix}414\\548\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{19}&\frac{4}{19}\\\frac{5}{38}&-\frac{7}{76}\end{matrix}\right)\left(\begin{matrix}414\\548\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{19}\times 414+\frac{4}{19}\times 548\\\frac{5}{38}\times 414-\frac{7}{76}\times 548\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}50\\4\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=50,y=4$$

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.

$$7x+16y=414,10x+12y=548$$

To make 7x and 10x equal, multiply all terms on each side of the first equation by 10 and all terms on each side of the second by 7.

$$10\times 7x+10\times 16y=10\times 414,7\times 10x+7\times 12y=7\times 548$$

Simplify.

$$70x+160y=4140,70x+84y=3836$$

Subtract 70x+84y=3836 from 70x+160y=4140 by subtracting like terms on each side of the equal sign.

$$70x-70x+160y-84y=4140-3836$$

Add 70x to -70x. Terms 70x and -70x cancel out, leaving an equation with only one variable that can be solved.

$$160y-84y=4140-3836$$

Add 160y to -84y.

$$76y=4140-3836$$

Add 4140 to -3836.

$$76y=304$$

Divide both sides by 76.

$$y=4$$

Substitute 4 for y in 10x+12y=548. Because the resulting equation contains only one variable, you can solve for x directly.

$$10x+12\times 4=548$$

Multiply 12 times 4.

$$10x+48=548$$

Subtract 48 from both sides of the equation.

$$10x=500$$

Divide both sides by 10.

$$x=50$$

The system is now solved.

$$x=50,y=4$$