Solve S=frac{a_{1}+a_{9}+8d}{2}*17

$$S = \frac { a _ { 1 } + a _ { 9 } + 8 d } { 2 } \cdot 17$$

Solve for S

$$S=\frac{17\left(a_{1}+a_{9}+8d\right)}{2}$$

Solution Steps

Express \frac{a_{1}+a_{9}+8d}{2}\times 17 as a single fraction.

$$S=\frac{\left(a_{1}+a_{9}+8d\right)\times 17}{2}$$

Use the distributive property to multiply a_{1}+a_{9}+8d by 17.

$$S=\frac{17a_{1}+17a_{9}+136d}{2}$$

Divide each term of 17a_{1}+17a_{9}+136d by 2 to get \frac{17}{2}a_{1}+\frac{17}{2}a_{9}+68d.

$$S=\frac{17}{2}a_{1}+\frac{17}{2}a_{9}+68d$$

Solve for a_1

$$a_{1}=\frac{2S}{17}-a_{9}-8d$$

Steps for Solving Linear Equation

Express \frac{a_{1}+a_{9}+8d}{2}\times 17 as a single fraction.

$$S=\frac{\left(a_{1}+a_{9}+8d\right)\times 17}{2}$$

Use the distributive property to multiply a_{1}+a_{9}+8d by 17.

$$S=\frac{17a_{1}+17a_{9}+136d}{2}$$

Divide each term of 17a_{1}+17a_{9}+136d by 2 to get \frac{17}{2}a_{1}+\frac{17}{2}a_{9}+68d.

$$S=\frac{17}{2}a_{1}+\frac{17}{2}a_{9}+68d$$

Swap sides so that all variable terms are on the left hand side.

$$\frac{17}{2}a_{1}+\frac{17}{2}a_{9}+68d=S$$

Subtract \frac{17}{2}a_{9} from both sides.

$$\frac{17}{2}a_{1}+68d=S-\frac{17}{2}a_{9}$$

Subtract 68d from both sides.

$$\frac{17}{2}a_{1}=S-\frac{17}{2}a_{9}-68d$$

The equation is in standard form.

$$\frac{17}{2}a_{1}=-\frac{17a_{9}}{2}+S-68d$$

Divide both sides of the equation by \frac{17}{2}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{17}{2}a_{1}}{\frac{17}{2}}=\frac{-\frac{17a_{9}}{2}+S-68d}{\frac{17}{2}}$$

Dividing by \frac{17}{2} undoes the multiplication by \frac{17}{2}.

$$a_{1}=\frac{-\frac{17a_{9}}{2}+S-68d}{\frac{17}{2}}$$

Divide S-\frac{17a_{9}}{2}-68d by \frac{17}{2} by multiplying S-\frac{17a_{9}}{2}-68d by the reciprocal of \frac{17}{2}.

$$a_{1}=\frac{2S}{17}-a_{9}-8d$$