Answer:
[tex] x = \dfrac{118}{27}[/tex]
Step-by-step explanation:
Let's denote the linear functions as follows:
- [tex] f(x) [/tex] is the original linear function, and its equation is [tex] y = f(x) [/tex].
- [tex] g(x) [/tex] is the linear function with a slope 6 times that of [tex] f(x) [/tex], and its equation is [tex] y = g(x) [/tex].
Given that [tex] f(x) [/tex] passes through the points [tex](0, 3)[/tex] and [tex](2, 4)[/tex], we can find the slope of [tex] f(x) [/tex] using the formula for the slope ([tex]m[/tex]):
[tex] m = \dfrac{{\textsf{change in } y}}{{\textsf{change in } x}} [/tex]
For [tex] f(x) [/tex], the slope ([tex]m_f[/tex]) is given by:
[tex] m_f = \dfrac{{4 - 3}}{{2 - 0}} \\\\ = \dfrac{1}{2} [/tex]
Now, the slope of [tex] g(x) [/tex] is 6 times the slope of [tex] f(x) [/tex], so the slope of [tex] g(x) [/tex] ([tex]m_g[/tex]) is:
[tex] m_g = 6 \times \dfrac{1}{2} = 3 [/tex]
Now, we have the slope ([tex]m_g = 3[/tex]) of [tex] g(x) [/tex]. We are also given that the graph of [tex] g [/tex] passes through the point [tex](-8, -35)[/tex].
We can use the point-slope form of a linear equation to write the equation for [tex] g(x) [/tex]:
[tex] y - y_1 = m_g \times (x - x_1) [/tex]
Substitute the values into the equation:
[tex] y - (-35) = 3 \times (x - (-8)) [/tex]
[tex] y + 35 = 3 \times (x + 8) [/tex]
Now, simplify the equation:
[tex] y + 35 = 3x + 24 [/tex]
[tex] y = 3x - 11 [/tex]
Now, we want to find the value of [tex] x [/tex] for which [tex] f(x) = a [/tex], and given that [tex] 9(a) = 19 [/tex], we can write [tex] a = \dfrac{19}{9} [/tex].
So, set [tex] f(x) [/tex] equal to [tex] a [/tex]:
[tex] f(x) = a [/tex]
[tex] f(x) = \dfrac{19}{9} [/tex]
Now, we can find the value of [tex] x [/tex] by substituting [tex] f(x) = \dfrac{19}{9} [/tex] into the equation for [tex] f(x) [/tex]:
[tex] 3x - 11 = \dfrac{19}{9} [/tex]
Solve this equation for [tex] x [/tex]: [tex] 3x = \dfrac{19}{9}+11 [/tex]
[tex] 3x = \dfrac{118}{9}[/tex]
[tex] x =\dfrac{118}{9} \cdot \dfrac{1}{3} [/tex]
[tex] x = \dfrac{118}{27}[/tex]
So, the value of x that satisfies [tex] f(x) = a [/tex] is: [tex] \dfrac{118}{27}[/tex]
