Solve for y₁ d = square root of (x_2-x_1)^2+(y_2-y_1)^2

$d = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$

Rewrite the equation as $\sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}} = d$ .

$\sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}} = d$

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}}^{2} = d^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$ as ${({(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2})}^{\frac{1}{2}}$ .

${({({(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2})}^{\frac{1}{2}})}^{2} = d^{2}$

Simplify the left side.

Simplify ${({({(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2})}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({({(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2})}^{\frac{1}{2}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${({(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2})}^{\frac{1}{2} \cdot 2} = d^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${({(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2})}^{\frac{1}{2} \cdot 2} = d^{2}$

Rewrite the expression.

${({(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2})}^{1} = d^{2}$

Simplify.

${(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2} = d^{2}$

Solve for $y_{1}$ .

Subtract ${(x_{2} - x_{1})}^{2}$ from both sides of the equation.

${(y_{2} - y_{1})}^{2} = d^{2} - {(x_{2} - x_{1})}^{2}$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$y_{2} - y_{1} = \pm \sqrt{d^{2} - {(x_{2} - x_{1})}^{2}}$

Simplify $\pm \sqrt{d^{2} - {(x_{2} - x_{1})}^{2}}$ .

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = d$ and $b = x_{2} - x_{1}$ .

$y_{2} - y_{1} = \pm \sqrt{(d + x_{2} - x_{1}) (d - (x_{2} - x_{1}))}$

Simplify.

Apply the distributive property.

$y_{2} - y_{1} = \pm \sqrt{(d + x_{2} - x_{1}) (d - x_{2} - - x_{1})}$

Multiply $- - x_{1}$ .

Multiply $- 1$ by $- 1$ .

$y_{2} - y_{1} = \pm \sqrt{(d + x_{2} - x_{1}) (d - x_{2} + 1 x_{1})}$

Multiply $x_{1}$ by $1$ .

$y_{2} - y_{1} = \pm \sqrt{(d + x_{2} - x_{1}) (d - x_{2} + x_{1})}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$y_{2} - y_{1} = \sqrt{(d + x_{2} - x_{1}) (d - x_{2} + x_{1})}$

Subtract $y_{2}$ from both sides of the equation.

$- y_{1} = \sqrt{(d + x_{2} - x_{1}) (d - x_{2} + x_{1})} - y_{2}$

Divide each term in $- y_{1} = \sqrt{(d + x_{2} - x_{1}) (d - x_{2} + x_{1})} - y_{2}$ by $- 1$ and simplify.

Divide each term in $- y_{1} = \sqrt{(d + x_{2} - x_{1}) (d - x_{2} + x_{1})} - y_{2}$ by $- 1$ .

$\frac{- y_{1}}{- 1} = \frac{\sqrt{(d + x_{2} - x_{1}) (d - x_{2} + x_{1})}}{- 1} + \frac{- y_{2}}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{y_{1}}{1} = \frac{\sqrt{(d + x_{2} - x_{1}) (d - x_{2} + x_{1})}}{- 1} + \frac{- y_{2}}{- 1}$

Divide $y_{1}$ by $1$ .

$y_{1} = \frac{\sqrt{(d + x_{2} - x_{1}) (d - x_{2} + x_{1})}}{- 1} + \frac{- y_{2}}{- 1}$

Simplify the right side.

Simplify each term.

Move the negative one from the denominator of $\frac{\sqrt{(d + x_{2} - x_{1}) (d - x_{2} + x_{1})}}{- 1}$ .

$y_{1} = - 1 \cdot \sqrt{(d + x_{2} - x_{1}) (d - x_{2} + x_{1})} + \frac{- y_{2}}{- 1}$

Rewrite $- 1 \cdot \sqrt{(d + x_{2} - x_{1}) (d - x_{2} + x_{1})}$ as $- \sqrt{(d + x_{2} - x_{1}) (d - x_{2} + x_{1})}$ .

$y_{1} = - \sqrt{(d + x_{2} - x_{1}) (d - x_{2} + x_{1})} + \frac{- y_{2}}{- 1}$

Dividing two negative values results in a positive value.

$y_{1} = - \sqrt{(d + x_{2} - x_{1}) (d - x_{2} + x_{1})} + \frac{y_{2}}{1}$

Divide $y_{2}$ by $1$ .

$y_{1} = - \sqrt{(d + x_{2} - x_{1}) (d - x_{2} + x_{1})} + y_{2}$

Next, use the negative value of the $\pm$ to find the second solution.

$y_{2} - y_{1} = - \sqrt{(d + x_{2} - x_{1}) (d - x_{2} + x_{1})}$

Subtract $y_{2}$ from both sides of the equation.

$- y_{1} = - \sqrt{(d + x_{2} - x_{1}) (d - x_{2} + x_{1})} - y_{2}$

Divide each term in $- y_{1} = - \sqrt{(d + x_{2} - x_{1}) (d - x_{2} + x_{1})} - y_{2}$ by $- 1$ and simplify.

Divide each term in $- y_{1} = - \sqrt{(d + x_{2} - x_{1}) (d - x_{2} + x_{1})} - y_{2}$ by $- 1$ .

$\frac{- y_{1}}{- 1} = \frac{- \sqrt{(d + x_{2} - x_{1}) (d - x_{2} + x_{1})}}{- 1} + \frac{- y_{2}}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{y_{1}}{1} = \frac{- \sqrt{(d + x_{2} - x_{1}) (d - x_{2} + x_{1})}}{- 1} + \frac{- y_{2}}{- 1}$

Divide $y_{1}$ by $1$ .

$y_{1} = \frac{- \sqrt{(d + x_{2} - x_{1}) (d - x_{2} + x_{1})}}{- 1} + \frac{- y_{2}}{- 1}$

Simplify the right side.

Simplify each term.

Dividing two negative values results in a positive value.

$y_{1} = \frac{\sqrt{(d + x_{2} - x_{1}) (d - x_{2} + x_{1})}}{1} + \frac{- y_{2}}{- 1}$

Divide $\sqrt{(d + x_{2} - x_{1}) (d - x_{2} + x_{1})}$ by $1$ .

$y_{1} = \sqrt{(d + x_{2} - x_{1}) (d - x_{2} + x_{1})} + \frac{- y_{2}}{- 1}$

Dividing two negative values results in a positive value.

$y_{1} = \sqrt{(d + x_{2} - x_{1}) (d - x_{2} + x_{1})} + \frac{y_{2}}{1}$

Divide $y_{2}$ by $1$ .

$y_{1} = \sqrt{(d + x_{2} - x_{1}) (d - x_{2} + x_{1})} + y_{2}$

The complete solution is the result of both the positive and negative portions of the solution.

$y_{1} = - \sqrt{(d + x_{2} - x_{1}) (d - x_{2} + x_{1})} + y_{2}$

$y_{1} = \sqrt{(d + x_{2} - x_{1}) (d - x_{2} + x_{1})} + y_{2}$

$y_{1} = - \sqrt{(d + x_{2} - x_{1}) (d - x_{2} + x_{1})} + y_{2}$

$y_{1} = \sqrt{(d + x_{2} - x_{1}) (d - x_{2} + x_{1})} + y_{2}$

$y_{1} = - \sqrt{(d + x_{2} - x_{1}) (d - x_{2} + x_{1})} + y_{2}$

$y_{1} = \sqrt{(d + x_{2} - x_{1}) (d - x_{2} + x_{1})} + y_{2}$

Solve for y₁ d = square root of (x_2-x_1)^2+(y_2-y_1)^2

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