Solve {l}{-4(-1)^3+(a+2)(-1)^2-2(-1)-3b=0}{-4(2)^3+(a+2)(2)^2-2(2)-3b=9}

$$\left. \begin{array} { l } { - 4 ( - 1 ) ^ { 3 } + ( a + 2 ) ( - 1 ) ^ { 2 } - 2 ( - 1 ) - 3 b = 0 } \\ { - 4 ( 2 ) ^ { 3 } + ( a + 2 ) ( 2 ) ^ { 2 } - 2 ( 2 ) - 3 b = 9 } \end{array} \right.$$

Solve for a, b

$$a=15$$

$$b = \frac{23}{3} = 7\frac{2}{3} \approx 7.666666667$$

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.

$$a+2-3b+6=0,4\left(a+2\right)-3b-36=9$$

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

$$a+2-3b+6=0$$

Add 2 to 6.

$$a-3b+8=0$$

Subtract 8 from both sides of the equation.

$$a-3b=-8$$

Add 3b to both sides of the equation.

$$a=3b-8$$

Substitute 3b-8 for a in the other equation, 4\left(a+2\right)-3b-36=9.

$$4\left(3b-8+2\right)-3b-36=9$$

Add -8 to 2.

$$4\left(3b-6\right)-3b-36=9$$

Multiply 4 times -6+3b.

$$12b-24-3b-36=9$$

Add 12b to -3b.

$$9b-24-36=9$$

Add -24 to -36.

$$9b-60=9$$

Add 60 to both sides of the equation.

$$9b=69$$

Divide both sides by 9.

$$b=\frac{23}{3}$$

Substitute \frac{23}{3} for b in a=3b-8. Because the resulting equation contains only one variable, you can solve for a directly.

$$a=3\times \frac{23}{3}-8$$

Multiply 3 times \frac{23}{3}.

$$a=23-8$$

Add -8 to 23.

$$a=15$$

The system is now solved.

$$a=15,b=\frac{23}{3}$$

Steps Using Matrices

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

$$a+2-3b+6=0,4\left(a+2\right)-3b-36=9$$

Simplify the first equation to put it in standard form.

$$a+2-3b+6=0$$

Add 2 to 6.

$$a-3b+8=0$$

Subtract 8 from both sides of the equation.

$$a-3b=-8$$

Simplify the second equation to put it in standard form.

$$4\left(a+2\right)-3b-36=9$$

Multiply 4 times a+2.

$$4a+8-3b-36=9$$

Add 8 to -36.

$$4a-3b-28=9$$

Add 28 to both sides of the equation.

$$4a-3b=37$$

Write the equations in matrix form.

$$\left(\begin{matrix}1&-3\\4&-3\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-8\\37\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&-3\\4&-3\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-3\\4&-3\end{matrix}\right))\left(\begin{matrix}1&-3\\4&-3\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\4&-3\end{matrix}\right))\left(\begin{matrix}-8\\37\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\4&-3\end{matrix}\right))\left(\begin{matrix}-8\\37\end{matrix}\right)$$

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

$$\left(\begin{matrix}a\\b\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\4&-3\end{matrix}\right))\left(\begin{matrix}-8\\37\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{-3-\left(-3\times 4\right)}&-\frac{-3}{-3-\left(-3\times 4\right)}\\-\frac{4}{-3-\left(-3\times 4\right)}&\frac{1}{-3-\left(-3\times 4\right)}\end{matrix}\right)\left(\begin{matrix}-8\\37\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}&\frac{1}{3}\\-\frac{4}{9}&\frac{1}{9}\end{matrix}\right)\left(\begin{matrix}-8\\37\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}\left(-8\right)+\frac{1}{3}\times 37\\-\frac{4}{9}\left(-8\right)+\frac{1}{9}\times 37\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}a\\b\end{matrix}\right)=\left(\begin{matrix}15\\\frac{23}{3}\end{matrix}\right)$$

Extract the matrix elements a and b.

$$a=15,b=\frac{23}{3}$$