Solve {l}{7a-10b=-64}{5b+3a=19}

$$\left\{ \begin{array} { l } { 7 a - 10 b = - 64 } \\ { 5 b + 3 a = 19 } \end{array} \right.$$

Solve for a, b

$$a=-2$$

$$b=5$$

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.

$$7a-10b=-64,3a+5b=19$$

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

$$7a-10b=-64$$

Add 10b to both sides of the equation.

$$7a=10b-64$$

Divide both sides by 7.

$$a=\frac{1}{7}\left(10b-64\right)$$

Multiply \frac{1}{7} times 10b-64.

$$a=\frac{10}{7}b-\frac{64}{7}$$

Substitute \frac{10b-64}{7} for a in the other equation, 3a+5b=19.

$$3\left(\frac{10}{7}b-\frac{64}{7}\right)+5b=19$$

Multiply 3 times \frac{10b-64}{7}.

$$\frac{30}{7}b-\frac{192}{7}+5b=19$$

Add \frac{30b}{7} to 5b.

$$\frac{65}{7}b-\frac{192}{7}=19$$

Add \frac{192}{7} to both sides of the equation.

$$\frac{65}{7}b=\frac{325}{7}$$

Divide both sides of the equation by \frac{65}{7}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$b=5$$

Substitute 5 for b in a=\frac{10}{7}b-\frac{64}{7}. Because the resulting equation contains only one variable, you can solve for a directly.

$$a=\frac{10}{7}\times 5-\frac{64}{7}$$

Multiply \frac{10}{7} times 5.

$$a=\frac{50-64}{7}$$

Add -\frac{64}{7} to \frac{50}{7} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a=-2$$

The system is now solved.

$$a=-2,b=5$$

Steps Using Matrices

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

$$7a-10b=-64,3a+5b=19$$

Write the equations in matrix form.

$$\left(\begin{matrix}7&-10\\3&5\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-64\\19\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}7&-10\\3&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}7&-10\\3&5\end{matrix}\right))\left(\begin{matrix}7&-10\\3&5\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}7&-10\\3&5\end{matrix}\right))\left(\begin{matrix}-64\\19\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}7&-10\\3&5\end{matrix}\right))\left(\begin{matrix}-64\\19\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}7&-10\\3&5\end{matrix}\right))\left(\begin{matrix}-64\\19\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{5}{7\times 5-\left(-10\times 3\right)}&-\frac{-10}{7\times 5-\left(-10\times 3\right)}\\-\frac{3}{7\times 5-\left(-10\times 3\right)}&\frac{7}{7\times 5-\left(-10\times 3\right)}\end{matrix}\right)\left(\begin{matrix}-64\\19\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{13}&\frac{2}{13}\\-\frac{3}{65}&\frac{7}{65}\end{matrix}\right)\left(\begin{matrix}-64\\19\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{13}\left(-64\right)+\frac{2}{13}\times 19\\-\frac{3}{65}\left(-64\right)+\frac{7}{65}\times 19\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-2\\5\end{matrix}\right)$$

Extract the matrix elements a and b.

$$a=-2,b=5$$

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.

$$7a-10b=-64,3a+5b=19$$

To make 7a and 3a equal, multiply all terms on each side of the first equation by 3 and all terms on each side of the second by 7.

$$3\times 7a+3\left(-10\right)b=3\left(-64\right),7\times 3a+7\times 5b=7\times 19$$

Simplify.

$$21a-30b=-192,21a+35b=133$$

Subtract 21a+35b=133 from 21a-30b=-192 by subtracting like terms on each side of the equal sign.

$$21a-21a-30b-35b=-192-133$$

Add 21a to -21a. Terms 21a and -21a cancel out, leaving an equation with only one variable that can be solved.

$$-30b-35b=-192-133$$

Add -30b to -35b.

$$-65b=-192-133$$

Add -192 to -133.

$$-65b=-325$$

Divide both sides by -65.

$$b=5$$

Substitute 5 for b in 3a+5b=19. Because the resulting equation contains only one variable, you can solve for a directly.

$$3a+5\times 5=19$$

Multiply 5 times 5.

$$3a+25=19$$

Subtract 25 from both sides of the equation.

$$3a=-6$$

Divide both sides by 3.

$$a=-2$$

The system is now solved.

$$a=-2,b=5$$