Question

How many solutions does the system of equations x − y = 7 and y equals the square root of the quantity 3 times x plus 3 end quantity minus 2 have?

Answer 1

9514 1404 393

Answer:

  one solution (11, 4)

Step-by-step explanation:

Both equations describe curves with positive slope. The y-intercept of the line (-7) is less than that of the root function (-2), so the line must intersect the root function in exactly one place. There is one solution.

__

Solved in the conventional way, the equation would become a quadratic, with two solutions. One of them is extraneous. (That is why I prefer a graphical solution.)

Given ...

  x -y = 7

  y = √(3x +3) -2

We can substitute for y to get ...

  x -7 = √(3x +3) -2

  x -5 = √(3x +3)

  x^2 -10x +25 = 3x +3 . . . . . . square both sides

  x^2 -13x +22 = 0 = (x -11)(x -2) . . . . . gives solutions x=11, x=2

The value x=2 does not satisfy the equation ...

  x -7 = √(3x +3) -2   ⇒   2 -7 = √(3·2 +3) -2   ⇒   -5 = 1 (false)