Part 1: When writing linear equations, how do you determine which form of a line to use?
Part 2: Choose 1 set of points from the choices below. Then, solve the problem and post your solution, showing your steps.
Write an equation in point-slope form for the line that passes through one of the following pairs of points (you may choose the pair you want to work with). Then, use the same set of points to write the equation in standard form and again in slope-intercept form.
Point pairs
(5, 1), (–3, 4)
(0, –2), (3, 2)
(–2, –1), (1, 2)
It does not matter to me what form.
However if they give me two points like
(5, 1), (–3, 4)
, I am likely to do the following:
(y-1)/(x-5) = (4-1)/(-3-5)
(y-1)/(x-5) = (3)/(-8)
-8(y-1) = 3 (x-5)
-8y + 8 = 3 x -15
-8 y = 3 x -23
y = -(3/8) x + 23/8
which is in slope intercept form
but I could have done this:
m = (4-1)/(-3-5) = -3/8
so y = -(3/8) x + b
1 = -(3/8)5 + b
1 = -15/8 + b
b = 8/8+15/8 = 23/8
so
y = -(3/8) x + 23/8
which is remarkably similar to what I got the first way.
of course I could write that as
8 y = -3 x + 23
or
3 x + 8 y = 23
or whatever
Part 1: When writing linear equations, there are different forms that can be used, including slope-intercept form, point-slope form, and standard form. The choice of which form to use depends on the information you have about the line.
1. Slope-Intercept Form (y = mx + b): This form is useful when you know the slope (m) of the line and the y-intercept (b), which is the point where the line crosses the y-axis. If you have these values, you can directly write the equation in this form.
2. Point-Slope Form (y - y₁ = m(x - x₁)): This form is useful when you have the slope (m) of the line and the coordinates (x₁, y₁) of a point on the line. You can substitute these values into the equation to get the equation of the line.
3. Standard Form (Ax + By = C): This form is useful when you have the coefficients (A, B, and C) of the equation. The values of A, B, and C are usually integers, and A should not be negative. To convert an equation from another form to standard form, you may need to perform some algebraic manipulations.
Part 2: Let's choose the point pair (0, -2) and (3, 2) to work with.
To find the equation in point-slope form, we need the slope (m) and the coordinates of a point (x₁, y₁) on the line. We can use the formula:
m = (y₂ - y₁) / (x₂ - x₁)
Let's calculate the slope using the points (0, -2) and (3, 2):
m = (2 - (-2)) / (3 - 0) = 4 / 3
Now, we can substitute the slope and the point (0, -2) into the point-slope form equation:
y - y₁ = m(x - x₁)
y - (-2) = (4/3)(x - 0)
y + 2 = (4/3)x
To write the equation in standard form, we need to rearrange the point-slope form equation:
y + 2 = (4/3)x
3(y + 2) = 4x
3y + 6 = 4x
4x - 3y = -6
Finally, let's write the equation in slope-intercept form by solving for y:
y + 2 = (4/3)x
y = (4/3)x - 2
So, the equation for the line passing through (0, -2) and (3, 2) in each form is:
Point-Slope Form: y + 2 = (4/3)x
Standard Form: 4x - 3y = -6
Slope-Intercept Form: y = (4/3)x - 2