Question

Solve the system by substitution.

y=-7x^2-9x+6
y=1/2x+11

•no solution
•(0,11)
•(0,11), (2,12)
•(-4,9),(10,16)

Answer 1

A. no solution

Explanation:

We can use the second expression to substitute for y in the first expression. Then we have ...

1/2x +11 = -7x^2 -9x +6

Subtracting the right side, we get

7x^2 +(9 1/2)x +5 = 0

Eliminating fractions by multiplying by 2, this is ...

14x^2 +19x +10 = 0

The discriminant of this equation will tell the number and kind of roots. For a=14, b=19, c=10, the discriminant is ...

b^ -4ac = 19^2 -4·14·10 = 361 - 560 = -199

Since this value is negative, we know the two roots to the quadratic will be complex. There are no real solutions.

The graph in the attachment shows the two curves do not intersect, hence there are no values of x that will make the y-values match.

Answer 2

• no solution

Explanation:

We can use the second expression to substitute for y in the first expression. Then we have ...

... 1/2x +11 = -7x^2 -9x +6

Subtracting the right side, we get

... 7x^2 +(9 1/2)x +5 = 0

Eliminating fractions by multiplying by 2, this is ...

... 14x^2 +19x +10 = 0

The discriminant of this equation will tell the number and kind of roots. For a=14, b=19, c=10, the discriminant is ...

... b^ -4ac = 19^2 -4·14·10 = 361 - 560 = -199

Since this value is negative, we know the two roots to the quadratic will be complex. There are no real solutions.

_____

The graph shows the two curves do not intersect, hence there are no values of x that will make the y-values match.