Math Snap

STEP 1

1. We are given definite integrals of $f(x)$ and $g(x)$ over different intervals.
2. We need to evaluate a linear combination of these functions over the interval $[0, 2]$.

STEP 2

1. Use the properties of definite integrals to separate the given integral.
2. Substitute the known values of the integrals.
3. Calculate the result.

STEP 3

Use the linearity of integrals to separate the given integral:
$$\int_{0}^{2} (7f(x) - 6g(x)) \, dx = \int_{0}^{2} 7f(x) \, dx - \int_{0}^{2} 6g(x) \, dx$$ Apply the constant multiple rule to move constants outside the integrals:
$$= 7 \int_{0}^{2} f(x) \, dx - 6 \int_{0}^{2} g(x) \, dx$$

STEP 4

Substitute the known values of the integrals:
$$= 7(-9) - 6(3)$$

Calculate the result:
$$= -63 - 18$$ $$= -81$$ The value of the integral is:
$$\boxed{-81}$$