Solve a_{1}+5d=1/2(a_{1}+7d)+1

$$a _ { 1 } + 5 d = \frac { 1 } { 2 } ( a _ { 1 } + 7 d ) + 1$$

Solve for a_1

$$a_{1}=2-3d$$

Steps for Solving Linear Equation

Use the distributive property to multiply \frac{1}{2} by a_{1}+7d.

$$a_{1}+5d=\frac{1}{2}a_{1}+\frac{7}{2}d+1$$

Subtract \frac{1}{2}a_{1} from both sides.

$$a_{1}+5d-\frac{1}{2}a_{1}=\frac{7}{2}d+1$$

Combine a_{1} and -\frac{1}{2}a_{1} to get \frac{1}{2}a_{1}.

$$\frac{1}{2}a_{1}+5d=\frac{7}{2}d+1$$

Subtract 5d from both sides.

$$\frac{1}{2}a_{1}=\frac{7}{2}d+1-5d$$

Combine \frac{7}{2}d and -5d to get -\frac{3}{2}d.

$$\frac{1}{2}a_{1}=-\frac{3}{2}d+1$$

The equation is in standard form.

$$\frac{1}{2}a_{1}=-\frac{3d}{2}+1$$

Multiply both sides by 2.

$$\frac{\frac{1}{2}a_{1}}{\frac{1}{2}}=\frac{-\frac{3d}{2}+1}{\frac{1}{2}}$$

Dividing by \frac{1}{2} undoes the multiplication by \frac{1}{2}.

$$a_{1}=\frac{-\frac{3d}{2}+1}{\frac{1}{2}}$$

Divide -\frac{3d}{2}+1 by \frac{1}{2} by multiplying -\frac{3d}{2}+1 by the reciprocal of \frac{1}{2}.

$$a_{1}=2-3d$$

Solve for d

$$d=\frac{2-a_{1}}{3}$$

Steps for Solving Linear Equation

Use the distributive property to multiply \frac{1}{2} by a_{1}+7d.

$$a_{1}+5d=\frac{1}{2}a_{1}+\frac{7}{2}d+1$$

Subtract \frac{7}{2}d from both sides.

$$a_{1}+5d-\frac{7}{2}d=\frac{1}{2}a_{1}+1$$

Combine 5d and -\frac{7}{2}d to get \frac{3}{2}d.

$$a_{1}+\frac{3}{2}d=\frac{1}{2}a_{1}+1$$

Subtract a_{1} from both sides.

$$\frac{3}{2}d=\frac{1}{2}a_{1}+1-a_{1}$$

Combine \frac{1}{2}a_{1} and -a_{1} to get -\frac{1}{2}a_{1}.

$$\frac{3}{2}d=-\frac{1}{2}a_{1}+1$$

The equation is in standard form.

$$\frac{3}{2}d=-\frac{a_{1}}{2}+1$$

Divide both sides of the equation by \frac{3}{2}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{3}{2}d}{\frac{3}{2}}=\frac{-\frac{a_{1}}{2}+1}{\frac{3}{2}}$$

Dividing by \frac{3}{2} undoes the multiplication by \frac{3}{2}.

$$d=\frac{-\frac{a_{1}}{2}+1}{\frac{3}{2}}$$

Divide -\frac{a_{1}}{2}+1 by \frac{3}{2} by multiplying -\frac{a_{1}}{2}+1 by the reciprocal of \frac{3}{2}.

$$d=\frac{2-a_{1}}{3}$$