The equation of a line given two points needs to be found. Samuel claims that slope-intercept form will generate the equation and Helena claims that point-slope form will find the equation. Who is correct? Explain your reason by describing both forms.

both will work

the slope of the line between the two points is (y2-y1)/(x2-x1). So, the point-slope form yields
y-y1 = (y2-y1)/(x2-x1) (x-x1)
But that is just the same as
y = (y2-y1)/(x2-x1) x - (y2-y1)/(x2-x1) x1 + y1

or,
y-y1 = m(x-x1)
is the same as
y = mx + (y1 - m*x1)
which is
y = mx+b where b = y1 - m*x1

Well, let me juggle with this question for a moment.

Both Samuel and Helena have a valid point here, no pun intended.

Starting with the slope-intercept form, it is given by the equation: y = mx + b. In this form, 'm' represents the slope of the line, and 'b' represents the y-intercept, where the line crosses the y-axis. If you know the slope and the y-intercept, you can plug those values into this equation and ta-da! There's your line equation.

On the other hand, we have the point-slope form, which is given by the equation: y - y₁ = m(x - x₁). Here, 'm' again represents the slope, and (x₁, y₁) represents a point on the line. This form allows you to use the coordinates of one point, along with the slope, to find the equation of the line.

So, who's correct? Well, it depends on what information you have at hand. If you already know the slope and y-intercept, then Samuel is right, and you can use the slope-intercept form to find the equation. However, if you only have the coordinates of a point and the slope, then Helena is correct, and you can use the point-slope form to find the equation.

In the end, they both have their merits, just like a clown juggling balls of different colors. Its all about which form is more convenient for the information you have.

Both Samuel and Helena are correct, as both slope-intercept form and point-slope form can be used to find the equation of a line given two points. However, they represent the equation in different forms.

1. Slope-Intercept Form:
The slope-intercept form of a linear equation is given by: y = mx + b, where m represents the slope of the line, and b represents the y-intercept. In this form, the equation is solved for y. To use the slope-intercept form to find the equation of a line, you typically need to know the slope of the line and the y-coordinate of one point on the line.

2. Point-Slope Form:
The point-slope form of a linear equation is given by: y - y1 = m(x - x1), where m represents the slope of the line, and (x1, y1) represents the coordinates of a point on the line. In this form, the equation is presented in terms of a specific point on the line and the slope of the line. To use the point-slope form to find the equation of a line, you need to know the coordinates of two points on the line.

So, Samuel is correct that slope-intercept form can be used to find the equation of a line given two points. However, you would need to compute the slope first and then substitute it into the equation along with the coordinates of one of the points to find the equation.

Helena is also correct that point-slope form can be used to find the equation of a line given two points. In this case, you would substitute the coordinates of one of the points into the equation, along with the slope calculated using the two points, to find the equation.

Both Samuel and Helena are correct, as both slope-intercept form and point-slope form can be used to generate the equation of a line given two points. Let's define and describe both forms:

1. Slope-Intercept Form: The equation of a line in slope-intercept form is represented as y = mx + b, where:
- y and x represent the coordinates of any point on the line
- m represents the slope of the line
- b represents the y-intercept of the line (the point where the line intersects the y-axis)

To find the equation of a line using slope-intercept form given two points (x₁, y₁) and (x₂, y₂), we can follow these steps:
- Calculate the slope (m) of the line using the formula m = (y₂ - y₁) / (x₂ - x₁).
- Once we have the slope, we can substitute any value (x, y) of one of the given points into the equation and solve for b.
- After determining the value of b, we can plug in the slope (m) and y-intercept (b) into the equation y = mx + b to get the final equation of the line.

2. Point-Slope Form: The equation of a line in point-slope form is represented as y - y₁ = m(x - x₁), where:
- y and x represent the coordinates of any point on the line
- m represents the slope of the line
- (x₁, y₁) represents the coordinates of a specific point on the line

To find the equation of a line using point-slope form given two points (x₁, y₁) and (x₂, y₂), we can follow these steps:
- Calculate the slope (m) of the line using the formula m = (y₂ - y₁) / (x₂ - x₁).
- Once we have the slope, we can substitute the coordinates of one of the given points (x₁, y₁) into the point-slope equation y - y₁ = m(x - x₁).
- After substituting the values, we can simplify and rearrange the equation to obtain the final equation of the line in point-slope form.

In summary, both forms (slope-intercept form and point-slope form) can be used to find the equation of a line given two points. It is ultimately a matter of personal preference or the requirements of the given problem as to which form is used.