Question

the distance between the points (10,y) and (4,8) is √61. arrange the equations below to show the process for finding the value of y.
√61 = √(10 - 4)²+(y - 8)²
√61 = √36+(y - 8)²
61 = 36+(y - 8)²
25=(y - 8)²
±5=y - 8
y = 13 or y = 3

Explanation:

Step1: Apply distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Here, $x_1 = 4,y_1 = 8,x_2=10,y_2 = y$ and $d=\sqrt{61}$. So we have $\sqrt{61}=\sqrt{(10 - 4)^2+(y - 8)^2}$.

Step2: Simplify the equation

$(10 - 4)^2=36$, so $\sqrt{61}=\sqrt{36+(y - 8)^2}$. Squaring both sides to get rid of the square - roots, we obtain $61=36+(y - 8)^2$.

Step3: Isolate the squared term

Subtract 36 from both sides: $61-36=(y - 8)^2$, which simplifies to $25=(y - 8)^2$.

Step4: Solve for $y - 8$

Taking the square root of both sides, we have $y - 8=\pm5$.

Step5: Solve for $y$

If $y - 8 = 5$, then $y=13$; if $y - 8=-5$, then $y = 3$.

Answer:

$y = 13$ or $y = 3$