Solve left(a-1right)^2+{left(7-3a/8+2right)}^2=left(a-4right)^2+{left(7-3a/8+3right)}^2

$${ \left(a-1 \right) }^{ 2 } + { \left( \frac{ 7-3a }{ 8 } +2 \right) }^{ 2 } = { \left(a-4 \right) }^{ 2 } + { \left( \frac{ 7-3a }{ 8 } +3 \right) }^{ 2 }$$

Solve for a

$$a = \frac{29}{9} = 3\frac{2}{9} \approx 3.222222222$$

Steps for Solving Linear Equation

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(a-1\right)^{2}.

$$a^{2}-2a+1+\left(\frac{7-3a}{8}+2\right)^{2}=\left(a-4\right)^{2}+\left(\frac{7-3a}{8}+3\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\frac{7-3a}{8}+2\right)^{2}.

$$a^{2}-2a+1+\left(\frac{7-3a}{8}\right)^{2}+4\times \frac{7-3a}{8}+4=\left(a-4\right)^{2}+\left(\frac{7-3a}{8}+3\right)^{2}$$

To raise \frac{7-3a}{8} to a power, raise both numerator and denominator to the power and then divide.

$$a^{2}-2a+1+\frac{\left(7-3a\right)^{2}}{8^{2}}+4\times \frac{7-3a}{8}+4=\left(a-4\right)^{2}+\left(\frac{7-3a}{8}+3\right)^{2}$$

Cancel out 8, the greatest common factor in 4 and 8.

$$a^{2}-2a+1+\frac{\left(7-3a\right)^{2}}{8^{2}}+\frac{7-3a}{2}+4=\left(a-4\right)^{2}+\left(\frac{7-3a}{8}+3\right)^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 8^{2} and 2 is 64. Multiply \frac{7-3a}{2} times \frac{32}{32}.

$$a^{2}-2a+1+\frac{\left(7-3a\right)^{2}}{64}+\frac{32\left(7-3a\right)}{64}+4=\left(a-4\right)^{2}+\left(\frac{7-3a}{8}+3\right)^{2}$$

Since \frac{\left(7-3a\right)^{2}}{64} and \frac{32\left(7-3a\right)}{64} have the same denominator, add them by adding their numerators.

$$a^{2}-2a+1+\frac{\left(7-3a\right)^{2}+32\left(7-3a\right)}{64}+4=\left(a-4\right)^{2}+\left(\frac{7-3a}{8}+3\right)^{2}$$

Do the multiplications in \left(7-3a\right)^{2}+32\left(7-3a\right).

$$a^{2}-2a+1+\frac{49-42a+9a^{2}+224-96a}{64}+4=\left(a-4\right)^{2}+\left(\frac{7-3a}{8}+3\right)^{2}$$

Combine like terms in 49-42a+9a^{2}+224-96a.

$$a^{2}-2a+1+\frac{273-138a+9a^{2}}{64}+4=\left(a-4\right)^{2}+\left(\frac{7-3a}{8}+3\right)^{2}$$

Add 1 and 4 to get 5.

$$a^{2}-2a+5+\frac{273-138a+9a^{2}}{64}=\left(a-4\right)^{2}+\left(\frac{7-3a}{8}+3\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(a-4\right)^{2}.

$$a^{2}-2a+5+\frac{273-138a+9a^{2}}{64}=a^{2}-8a+16+\left(\frac{7-3a}{8}+3\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\frac{7-3a}{8}+3\right)^{2}.

$$a^{2}-2a+5+\frac{273-138a+9a^{2}}{64}=a^{2}-8a+16+\left(\frac{7-3a}{8}\right)^{2}+6\times \frac{7-3a}{8}+9$$

To raise \frac{7-3a}{8} to a power, raise both numerator and denominator to the power and then divide.

$$a^{2}-2a+5+\frac{273-138a+9a^{2}}{64}=a^{2}-8a+16+\frac{\left(7-3a\right)^{2}}{8^{2}}+6\times \frac{7-3a}{8}+9$$

Express 6\times \frac{7-3a}{8} as a single fraction.

$$a^{2}-2a+5+\frac{273-138a+9a^{2}}{64}=a^{2}-8a+16+\frac{\left(7-3a\right)^{2}}{8^{2}}+\frac{6\left(7-3a\right)}{8}+9$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 8^{2} and 8 is 64. Multiply \frac{6\left(7-3a\right)}{8} times \frac{8}{8}.

$$a^{2}-2a+5+\frac{273-138a+9a^{2}}{64}=a^{2}-8a+16+\frac{\left(7-3a\right)^{2}}{64}+\frac{8\times 6\left(7-3a\right)}{64}+9$$

Since \frac{\left(7-3a\right)^{2}}{64} and \frac{8\times 6\left(7-3a\right)}{64} have the same denominator, add them by adding their numerators.

$$a^{2}-2a+5+\frac{273-138a+9a^{2}}{64}=a^{2}-8a+16+\frac{\left(7-3a\right)^{2}+8\times 6\left(7-3a\right)}{64}+9$$

Do the multiplications in \left(7-3a\right)^{2}+8\times 6\left(7-3a\right).

$$a^{2}-2a+5+\frac{273-138a+9a^{2}}{64}=a^{2}-8a+16+\frac{49-42a+9a^{2}+336-144a}{64}+9$$

Combine like terms in 49-42a+9a^{2}+336-144a.

$$a^{2}-2a+5+\frac{273-138a+9a^{2}}{64}=a^{2}-8a+16+\frac{385-186a+9a^{2}}{64}+9$$

Add 16 and 9 to get 25.

$$a^{2}-2a+5+\frac{273-138a+9a^{2}}{64}=a^{2}-8a+25+\frac{385-186a+9a^{2}}{64}$$

Divide each term of 273-138a+9a^{2} by 64 to get \frac{273}{64}-\frac{69}{32}a+\frac{9}{64}a^{2}.

$$a^{2}-2a+5+\frac{273}{64}-\frac{69}{32}a+\frac{9}{64}a^{2}=a^{2}-8a+25+\frac{385-186a+9a^{2}}{64}$$

Add 5 and \frac{273}{64} to get \frac{593}{64}.

$$a^{2}-2a+\frac{593}{64}-\frac{69}{32}a+\frac{9}{64}a^{2}=a^{2}-8a+25+\frac{385-186a+9a^{2}}{64}$$

Combine -2a and -\frac{69}{32}a to get -\frac{133}{32}a.

$$a^{2}-\frac{133}{32}a+\frac{593}{64}+\frac{9}{64}a^{2}=a^{2}-8a+25+\frac{385-186a+9a^{2}}{64}$$

Combine a^{2} and \frac{9}{64}a^{2} to get \frac{73}{64}a^{2}.

$$\frac{73}{64}a^{2}-\frac{133}{32}a+\frac{593}{64}=a^{2}-8a+25+\frac{385-186a+9a^{2}}{64}$$

Divide each term of 385-186a+9a^{2} by 64 to get \frac{385}{64}-\frac{93}{32}a+\frac{9}{64}a^{2}.

$$\frac{73}{64}a^{2}-\frac{133}{32}a+\frac{593}{64}=a^{2}-8a+25+\frac{385}{64}-\frac{93}{32}a+\frac{9}{64}a^{2}$$

Add 25 and \frac{385}{64} to get \frac{1985}{64}.

$$\frac{73}{64}a^{2}-\frac{133}{32}a+\frac{593}{64}=a^{2}-8a+\frac{1985}{64}-\frac{93}{32}a+\frac{9}{64}a^{2}$$

Combine -8a and -\frac{93}{32}a to get -\frac{349}{32}a.

$$\frac{73}{64}a^{2}-\frac{133}{32}a+\frac{593}{64}=a^{2}-\frac{349}{32}a+\frac{1985}{64}+\frac{9}{64}a^{2}$$

Combine a^{2} and \frac{9}{64}a^{2} to get \frac{73}{64}a^{2}.

$$\frac{73}{64}a^{2}-\frac{133}{32}a+\frac{593}{64}=\frac{73}{64}a^{2}-\frac{349}{32}a+\frac{1985}{64}$$

Subtract \frac{73}{64}a^{2} from both sides.

$$\frac{73}{64}a^{2}-\frac{133}{32}a+\frac{593}{64}-\frac{73}{64}a^{2}=-\frac{349}{32}a+\frac{1985}{64}$$

Combine \frac{73}{64}a^{2} and -\frac{73}{64}a^{2} to get 0.

$$-\frac{133}{32}a+\frac{593}{64}=-\frac{349}{32}a+\frac{1985}{64}$$

Add \frac{349}{32}a to both sides.

$$-\frac{133}{32}a+\frac{593}{64}+\frac{349}{32}a=\frac{1985}{64}$$

Combine -\frac{133}{32}a and \frac{349}{32}a to get \frac{27}{4}a.

$$\frac{27}{4}a+\frac{593}{64}=\frac{1985}{64}$$

Subtract \frac{593}{64} from both sides.

$$\frac{27}{4}a=\frac{1985}{64}-\frac{593}{64}$$

Subtract \frac{593}{64} from \frac{1985}{64} to get \frac{87}{4}.

$$\frac{27}{4}a=\frac{87}{4}$$

Multiply both sides by \frac{4}{27}, the reciprocal of \frac{27}{4}.

$$a=\frac{87}{4}\times \frac{4}{27}$$

Multiply \frac{87}{4} and \frac{4}{27} to get \frac{29}{9}.

$$a=\frac{29}{9}$$