Solve {l}{d=10,410+2.65g}{d=20g}

$$\left. \begin{array} { l } { d = 10,410 + 2.65 g } \\ { d = 20 g } \end{array} \right.$$

Solve for d, g

$$d=12000$$

$$g=600$$

Steps Using Substitution

Consider the first equation. Subtract 2.65g from both sides.

$$d-2.65g=10410$$

Consider the second equation. Subtract 20g from both sides.

$$d-20g=0$$

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.

$$d-2.65g=10410,d-20g=0$$

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

$$d-2.65g=10410$$

Add \frac{53g}{20} to both sides of the equation.

$$d=2.65g+10410$$

Substitute \frac{53g}{20}+10410 for d in the other equation, d-20g=0.

$$2.65g+10410-20g=0$$

Add \frac{53g}{20} to -20g.

$$-17.35g+10410=0$$

Subtract 10410 from both sides of the equation.

$$-17.35g=-10410$$

Divide both sides of the equation by -17.35, which is the same as multiplying both sides by the reciprocal of the fraction.

$$g=600$$

Substitute 600 for g in d=2.65g+10410. Because the resulting equation contains only one variable, you can solve for d directly.

$$d=2.65\times 600+10410$$

Multiply 2.65 times 600.

$$d=1590+10410$$

Add 10410 to 1590.

$$d=12000$$

The system is now solved.

$$d=12000,g=600$$

Steps Using Matrices

Consider the first equation. Subtract 2.65g from both sides.

$$d-2.65g=10410$$

Consider the second equation. Subtract 20g from both sides.

$$d-20g=0$$

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

$$d-2.65g=10410,d-20g=0$$

Write the equations in matrix form.

$$\left(\begin{matrix}1&-2.65\\1&-20\end{matrix}\right)\left(\begin{matrix}d\\g\end{matrix}\right)=\left(\begin{matrix}10410\\0\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&-2.65\\1&-20\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-2.65\\1&-20\end{matrix}\right))\left(\begin{matrix}1&-2.65\\1&-20\end{matrix}\right)\left(\begin{matrix}d\\g\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2.65\\1&-20\end{matrix}\right))\left(\begin{matrix}10410\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}d\\g\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2.65\\1&-20\end{matrix}\right))\left(\begin{matrix}10410\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}d\\g\end{matrix}\right)=inverse(\left(\begin{matrix}1&-2.65\\1&-20\end{matrix}\right))\left(\begin{matrix}10410\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}d\\g\end{matrix}\right)=\left(\begin{matrix}-\frac{20}{-20-\left(-2.65\right)}&-\frac{-2.65}{-20-\left(-2.65\right)}\\-\frac{1}{-20-\left(-2.65\right)}&\frac{1}{-20-\left(-2.65\right)}\end{matrix}\right)\left(\begin{matrix}10410\\0\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}d\\g\end{matrix}\right)=\left(\begin{matrix}\frac{400}{347}&-\frac{53}{347}\\\frac{20}{347}&-\frac{20}{347}\end{matrix}\right)\left(\begin{matrix}10410\\0\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}d\\g\end{matrix}\right)=\left(\begin{matrix}\frac{400}{347}\times 10410\\\frac{20}{347}\times 10410\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}d\\g\end{matrix}\right)=\left(\begin{matrix}12000\\600\end{matrix}\right)$$

Extract the matrix elements d and g.

$$d=12000,g=600$$

Steps Using Elimination

Consider the first equation. Subtract 2.65g from both sides.

$$d-2.65g=10410$$

Consider the second equation. Subtract 20g from both sides.

$$d-20g=0$$

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.

$$d-2.65g=10410,d-20g=0$$

Subtract d-20g=0 from d-2.65g=10410 by subtracting like terms on each side of the equal sign.

$$d-d-2.65g+20g=10410$$

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

$$-2.65g+20g=10410$$

Add -\frac{53g}{20} to 20g.

$$17.35g=10410$$

Divide both sides of the equation by 17.35, which is the same as multiplying both sides by the reciprocal of the fraction.

$$g=600$$

Substitute 600 for g in d-20g=0. Because the resulting equation contains only one variable, you can solve for d directly.

$$d-20\times 600=0$$

Multiply -20 times 600.

$$d-12000=0$$

Add 12000 to both sides of the equation.

$$d=12000$$

The system is now solved.

$$d=12000,g=600$$