Solve {l}{4v+7w=-22}{-3v+5w=-45}

$$\left. \begin{array} { l } { 4 v + 7 w = - 22 } \\ { - 3 v + 5 w = - 45 } \end{array} \right.$$

Solve for v, w

$$v=5$$

$$w=-6$$

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.

$$4v+7w=-22,-3v+5w=-45$$

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

$$4v+7w=-22$$

Subtract 7w from both sides of the equation.

$$4v=-7w-22$$

Divide both sides by 4.

$$v=\frac{1}{4}\left(-7w-22\right)$$

Multiply \frac{1}{4} times -7w-22.

$$v=-\frac{7}{4}w-\frac{11}{2}$$

Substitute -\frac{7w}{4}-\frac{11}{2} for v in the other equation, -3v+5w=-45.

$$-3\left(-\frac{7}{4}w-\frac{11}{2}\right)+5w=-45$$

Multiply -3 times -\frac{7w}{4}-\frac{11}{2}.

$$\frac{21}{4}w+\frac{33}{2}+5w=-45$$

Add \frac{21w}{4} to 5w.

$$\frac{41}{4}w+\frac{33}{2}=-45$$

Subtract \frac{33}{2} from both sides of the equation.

$$\frac{41}{4}w=-\frac{123}{2}$$

Divide both sides of the equation by \frac{41}{4}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$w=-6$$

Substitute -6 for w in v=-\frac{7}{4}w-\frac{11}{2}. Because the resulting equation contains only one variable, you can solve for v directly.

$$v=-\frac{7}{4}\left(-6\right)-\frac{11}{2}$$

Multiply -\frac{7}{4} times -6.

$$v=\frac{21-11}{2}$$

Add -\frac{11}{2} to \frac{21}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$v=5$$

The system is now solved.

$$v=5,w=-6$$

Steps Using Matrices

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

$$4v+7w=-22,-3v+5w=-45$$

Write the equations in matrix form.

$$\left(\begin{matrix}4&7\\-3&5\end{matrix}\right)\left(\begin{matrix}v\\w\end{matrix}\right)=\left(\begin{matrix}-22\\-45\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}4&7\\-3&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&7\\-3&5\end{matrix}\right))\left(\begin{matrix}4&7\\-3&5\end{matrix}\right)\left(\begin{matrix}v\\w\end{matrix}\right)=inverse(\left(\begin{matrix}4&7\\-3&5\end{matrix}\right))\left(\begin{matrix}-22\\-45\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}v\\w\end{matrix}\right)=inverse(\left(\begin{matrix}4&7\\-3&5\end{matrix}\right))\left(\begin{matrix}-22\\-45\end{matrix}\right)$$

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

$$\left(\begin{matrix}v\\w\end{matrix}\right)=inverse(\left(\begin{matrix}4&7\\-3&5\end{matrix}\right))\left(\begin{matrix}-22\\-45\end{matrix}\right)$$

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.

$$\left(\begin{matrix}v\\w\end{matrix}\right)=\left(\begin{matrix}\frac{5}{4\times 5-7\left(-3\right)}&-\frac{7}{4\times 5-7\left(-3\right)}\\-\frac{-3}{4\times 5-7\left(-3\right)}&\frac{4}{4\times 5-7\left(-3\right)}\end{matrix}\right)\left(\begin{matrix}-22\\-45\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}v\\w\end{matrix}\right)=\left(\begin{matrix}\frac{5}{41}&-\frac{7}{41}\\\frac{3}{41}&\frac{4}{41}\end{matrix}\right)\left(\begin{matrix}-22\\-45\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}v\\w\end{matrix}\right)=\left(\begin{matrix}\frac{5}{41}\left(-22\right)-\frac{7}{41}\left(-45\right)\\\frac{3}{41}\left(-22\right)+\frac{4}{41}\left(-45\right)\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}v\\w\end{matrix}\right)=\left(\begin{matrix}5\\-6\end{matrix}\right)$$

Extract the matrix elements v and w.

$$v=5,w=-6$$

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.

$$4v+7w=-22,-3v+5w=-45$$

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

$$-3\times 4v-3\times 7w=-3\left(-22\right),4\left(-3\right)v+4\times 5w=4\left(-45\right)$$

Simplify.

$$-12v-21w=66,-12v+20w=-180$$

Subtract -12v+20w=-180 from -12v-21w=66 by subtracting like terms on each side of the equal sign.

$$-12v+12v-21w-20w=66+180$$

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

$$-21w-20w=66+180$$

Add -21w to -20w.

$$-41w=66+180$$

Add 66 to 180.

$$-41w=246$$

Divide both sides by -41.

$$w=-6$$

Substitute -6 for w in -3v+5w=-45. Because the resulting equation contains only one variable, you can solve for v directly.

$$-3v+5\left(-6\right)=-45$$

Multiply 5 times -6.

$$-3v-30=-45$$

Add 30 to both sides of the equation.

$$-3v=-15$$

Divide both sides by -3.

$$v=5$$

The system is now solved.

$$v=5,w=-6$$