Find the value of $k$ such that $\dbinom{-2}{k}$ is a direction vector of the line with equation $y=\dfrac{4x+1}{7}$.
That’s incorrect
Ah, finding values! I haven't been this excited since I found a penny on the ground! So, to find the value of $k$, we need to determine the direction vector of the line. Let me grab my clown calculator and do some calculations!
The equation $y=\dfrac{4x+1}{7}$ is in slope-intercept form, which means the coefficient of $x$ is the slope. So, the slope of the line is $\frac{4}{7}$.
Now, let's think about what a direction vector is. It's like me giving you directions to the nearest clown circus, but instead of giving you step-by-step instructions, I just give you the general idea of where to go. A direction vector for a line tells you the general direction the line is going in.
In this case, the direction vector should have a slope of $\frac{4}{7}$. But here's the deal: $\binom{-2}{k}$ is not in slope-intercept form. The first component, $-2$, is the $x$-component of the vector, while the second component, $k$, is the $y$-component. So, we can write $\binom{-2}{k}$ as the vector $\langle -2, k \rangle$.
We want the slope of the line to be equal to $\frac{4}{7}$. So, we set up the equation:
$\frac{k}{-2} = \frac{4}{7}$
Cross-multiplying, we get:
$7k = -8$
Finally, we solve for $k$:
$k = \frac{-8}{7}$
So, the value of $k$ that makes $\binom{-2}{k}$ a direction vector of the line is $k = \frac{-8}{7}$. Hope I didn't make you too clownfused!
To find the value of $k$ such that $\binom{-2}{k}$ is a direction vector of the line, we need to compare the coefficients of $x$ and $y$ in the equation of the line with the components of the direction vector.
The equation of the line is given as $y=\frac{4x+1}{7}$. We can rewrite this equation in the standard form $Ax+By+C=0$ by multiplying through by $7$ to eliminate the fraction:
$$7y = 4x + 1$$
$$4x - 7y + 1 = 0.$$
The coefficients of $x$ and $y$ are $4$ and $-7$, respectively. Therefore, we want the direction vector $\binom{-2}{k}$ to have the same coefficients, meaning $k = -7$.
Hence, the value of $k$ that makes $\binom{-2}{k}$ a direction vector of the line $y=\frac{4x+1}{7}$ is $k=-7$.
type 24/7
Just type normally
e.g. 5/7 as a fraction