distance formula
distance between (x1,y1) and (x2,y2) is
[tex]D=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
so the disatnce between (1,4) and (5,y) is the same as the disatnce between (10,-3) and (5,y)
therefor
[tex]\sqrt{(5-1)^2+(y-4)^2}=\sqrt{(5-10)^2+(y-(-3))^2}[/tex]
solve for y
[tex]\sqrt{(4)^2+(y-4)^2}=\sqrt{(-5)^2+(y+3)^2}[/tex]
[tex]\sqrt{16+y^2-8y+16}=\sqrt{25+y^2+6y+9}[/tex]
[tex]\sqrt{y^2-8y+32}=\sqrt{y^2+6y+34}[/tex]
square both sides
[tex]y^2-8y+32=y^2+6y+34[/tex]
minus y² both sides
-8y+32=6y+34
add 8y both sides
32=14y+34
minus 34 both sides
-2=14y
divide both sides by 14
[tex]\frac{-1}{7}=y[/tex]
the point is [tex](5,\frac{-1}{7})[/tex]
