Question

Linear combination problem help?

Prove that
`f(x) = 12x^2 + 7x − 19`
is a linear combination of
`g(x) = 2x^2 + x − 3`
and
`h(x) = −3x^2 − 2x + 5`.

Answer 1

I have shown that the linear combination is:

`f(x) = 3g(x) + (-2)h(x)`

Explanation:

A linear combination is:

`f(x) = Ag(x) + Bh(x)`

Matching constant terms, the following must be true:

`A(-3)+B(5) = -19`

Move the coefficients to the front:

`-3A + 5B = -19" [1]"`

Matching linear terms, the following must be true:

`A(x)+B(-2x) = 7x`

Divide both sides of the equation by x:

`A+B(-2) = 7`

Move the coefficients to the front and mark it as equation [2]:

`A-2B = 7" [2]"`

Add 2B to both sides:

`A = 2B+7 " [2.1]"`

Substitute into equation [1]:

`-3(2B+7) + 5B = -19`

`-6B - 21 + 5B = -19`

`-B = 2`

`B = -2`

Use equation [2.1] to find the value of A:

`A = 2(-2)+7`

`A = 3`

Check:

`f(x) = 3g(x) + (-2)h(x)`

`f(x) = 3(2x^2 + x − 3) + (-2)(−3x^2 − 2x + 5)`

`f(x) = 6x^2 + 3x − 9 + 6x^2 + 4x -10`

`f(x) = 12x^2 + 7x − 19`

This checks.