Solve {l}{2a_{1}+d=20}{2a_{1}+5d=80}

$$\left\{ \begin{array} { l } { 2 a _ { 1 } + d = 20 } \\ { 2 a _ { 1 } + 5 d = 80 } \end{array} \right.$$

Solve for a_1, d

$$a_{1} = \frac{5}{2} = 2\frac{1}{2} = 2.5$$

$$d=15$$

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.

$$2a_{1}+d=20,2a_{1}+5d=80$$

Choose one of the equations and solve it for a_{1} by isolating a_{1} on the left hand side of the equal sign.

$$2a_{1}+d=20$$

Subtract d from both sides of the equation.

$$2a_{1}=-d+20$$

Divide both sides by 2.

$$a_{1}=\frac{1}{2}\left(-d+20\right)$$

Multiply \frac{1}{2} times -d+20.

$$a_{1}=-\frac{1}{2}d+10$$

Substitute -\frac{d}{2}+10 for a_{1} in the other equation, 2a_{1}+5d=80.

$$2\left(-\frac{1}{2}d+10\right)+5d=80$$

Multiply 2 times -\frac{d}{2}+10.

$$-d+20+5d=80$$

Add -d to 5d.

$$4d+20=80$$

Subtract 20 from both sides of the equation.

$$4d=60$$

Divide both sides by 4.

$$d=15$$

Substitute 15 for d in a_{1}=-\frac{1}{2}d+10. Because the resulting equation contains only one variable, you can solve for a_{1} directly.

$$a_{1}=-\frac{1}{2}\times 15+10$$

Multiply -\frac{1}{2} times 15.

$$a_{1}=-\frac{15}{2}+10$$

Add 10 to -\frac{15}{2}.

$$a_{1}=\frac{5}{2}$$

The system is now solved.

$$a_{1}=\frac{5}{2},d=15$$

Steps Using Matrices

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

$$2a_{1}+d=20,2a_{1}+5d=80$$

Write the equations in matrix form.

$$\left(\begin{matrix}2&1\\2&5\end{matrix}\right)\left(\begin{matrix}a_{1}\\d\end{matrix}\right)=\left(\begin{matrix}20\\80\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}2&1\\2&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&1\\2&5\end{matrix}\right))\left(\begin{matrix}2&1\\2&5\end{matrix}\right)\left(\begin{matrix}a_{1}\\d\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\2&5\end{matrix}\right))\left(\begin{matrix}20\\80\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}a_{1}\\d\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\2&5\end{matrix}\right))\left(\begin{matrix}20\\80\end{matrix}\right)$$

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

$$\left(\begin{matrix}a_{1}\\d\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\2&5\end{matrix}\right))\left(\begin{matrix}20\\80\end{matrix}\right)$$

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.

$$\left(\begin{matrix}a_{1}\\d\end{matrix}\right)=\left(\begin{matrix}\frac{5}{2\times 5-2}&-\frac{1}{2\times 5-2}\\-\frac{2}{2\times 5-2}&\frac{2}{2\times 5-2}\end{matrix}\right)\left(\begin{matrix}20\\80\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}a_{1}\\d\end{matrix}\right)=\left(\begin{matrix}\frac{5}{8}&-\frac{1}{8}\\-\frac{1}{4}&\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}20\\80\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}a_{1}\\d\end{matrix}\right)=\left(\begin{matrix}\frac{5}{8}\times 20-\frac{1}{8}\times 80\\-\frac{1}{4}\times 20+\frac{1}{4}\times 80\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}a_{1}\\d\end{matrix}\right)=\left(\begin{matrix}\frac{5}{2}\\15\end{matrix}\right)$$

Extract the matrix elements a_{1} and d.

$$a_{1}=\frac{5}{2},d=15$$

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.

$$2a_{1}+d=20,2a_{1}+5d=80$$

Subtract 2a_{1}+5d=80 from 2a_{1}+d=20 by subtracting like terms on each side of the equal sign.

$$2a_{1}-2a_{1}+d-5d=20-80$$

Add 2a_{1} to -2a_{1}. Terms 2a_{1} and -2a_{1} cancel out, leaving an equation with only one variable that can be solved.

$$d-5d=20-80$$

Add d to -5d.

$$-4d=20-80$$

Add 20 to -80.

$$-4d=-60$$

Divide both sides by -4.

$$d=15$$

Substitute 15 for d in 2a_{1}+5d=80. Because the resulting equation contains only one variable, you can solve for a_{1} directly.

$$2a_{1}+5\times 15=80$$

Multiply 5 times 15.

$$2a_{1}+75=80$$

Subtract 75 from both sides of the equation.

$$2a_{1}=5$$

Divide both sides by 2.

$$a_{1}=\frac{5}{2}$$

The system is now solved.

$$a_{1}=\frac{5}{2},d=15$$