Question

quiz instructions
practice test 4.1 is designed to prepare you for test 4.1
you may take this practice test as many times as you would like.
scratch paper is highly recommended.
practice tests and tests are 60% of your grade!
for non integer answers, write your answer as a reduced fraction
(for example, if you answer is \\(-\frac{2}{3}\\), type -2/3)

question 11
0.5 pts
write the equation of the line in slope intercept form using point slope formula: \\( y - y_1 = m (x - x_1) \\)
line with (-4, 0) and (-4, 6)

Explanation:

Step1: Calculate the slope \( m \)

The formula for slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Given points \((-4, 0)\) and \((-4, 6)\), we have \( x_1=-4,y_1 = 0,x_2=-4,y_2 = 6 \). Substituting into the slope formula: \( m=\frac{6 - 0}{-4-(-4)}=\frac{6}{0} \), which is undefined. This means the line is vertical.

Step2: Determine the equation of the line

A vertical line has the equation \( x = a \), where \( a \) is the x - coordinate of any point on the line. For the points \((-4, 0)\) and \((-4, 6)\), the x - coordinate is \(-4\). So the equation of the line is \( x=-4 \). But we need to check if we can express it in slope - intercept form (\( y=mx + b \)). Since the slope is undefined, the line cannot be written in the form \( y=mx + b \) (because \( m \) is undefined). However, if we consider the nature of the line, it is a vertical line at \( x = - 4 \), and for any \( y \) - value, \( x=-4 \). But the problem asks for slope - intercept form. Wait, a vertical line has an undefined slope, so it does not have a slope - intercept form in the traditional sense (\( y=mx + b \) requires \( m \) to be a real number). But let's re - examine the points. Both points have \( x=-4 \), so the line is \( x=-4 \), which can be thought of as a line where \( x \) is always \(-4\) regardless of \( y \). But if we try to use point - slope form: \( y - y_1=m(x - x_1) \), but since \( m \) is undefined, we can't use the point - slope form in the way we use it for non - vertical lines. The correct equation for the vertical line through \( x=-4 \) is \( x=-4 \). But the problem might have a typo or maybe we misinterpret. Wait, no, let's check again. The two points have the same \( x \) - coordinate (\( x=-4 \)), so it's a vertical line. The equation of a vertical line is \( x = - 4 \). Since the slope is undefined, we can't write it in \( y=mx + b \) form (because \( m \) doesn't exist). But maybe the problem expects us to recognize that it's a vertical line with equation \( x=-4 \), and since slope - intercept form is \( y = mx + b \), and this line can't be written in that form, but perhaps there's a mistake. Wait, no, let's check the points again: \((-4,0)\) and \((-4,6)\). The x - coordinates are the same, so it's a vertical line. So the equation is \( x=-4 \). But the problem says "Write the equation of the line in slope intercept form using point slope formula". Since the slope is undefined, we can't use the point - slope formula in the standard way. But maybe the problem has an error, or maybe we made a mistake. Wait, no, let's calculate the slope again. \( x_1=-4,y_1 = 0,x_2=-4,y_2 = 6 \). \( m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{6 - 0}{-4+4}=\frac{6}{0} \), which is undefined. So the line is vertical, \( x=-4 \).

Answer:

The line is a vertical line with equation \( x = - 4 \), and it cannot be written in slope - intercept form (\( y=mx + b \)) because the slope is undefined. However, if we consider the problem's requirement, and the fact that for a vertical line \( x=-4 \), we can state that the equation is \( x=-4 \). But if we strictly follow the slope - intercept form, since the slope is undefined, there is no slope - intercept form. But maybe the problem intended to have two points with different x - coordinates. Assuming that there was a typo and the points are, say, \((4,0)\) and \((-4,6)\), but based on the given points \((-4,0)\) and \((-4,6)\), the equation is \( x=-4 \). But since the problem asks for slope - intercept form, and the line is vertical, we can conclude that the line has the equation \( x=-4 \), and it does not have a slope - intercept form (as slope is undefined). But if we go back to the problem statement, maybe we misread the points. Wait, the points are \((-4,0)\) and \((-4,6)\). So the line is \( x=-4 \).