The first step of writing an equation when given two points is the same for both strategies, slope-intercept or point-slope forms. What would be the first step in finding the equation of the line that passes through (5, 1) and (3, 5)?
create a graph using the two points
find the slope
find the y-intercept
substitute one point’s coordinates into either form
find the slope
Use slope formula,m = StartFraction y 2 minus y 1 Over x 2 minus x 1 EndFraction, to find the slope of a line that passes through the points (–3, 8) and (4, –6).
m =
Use slope-intercept form, y = mx + b, to find the y-intercept (b) of the line.
b =
What is the new equation written in slope-intercept form, y = mx + b?
m = (8 - (-6))/( -3 - 4) = -2
b = y - mx = 8 - (-2)*(-3) = 2
The equation in slope-intercept form is y = -2x + 2.
Use slope-intercept form, y = mx + b to find the equation of the line that passes through the points (−6, 1) and (3, 4).
y = –3x + 5
y = 3x – 5
y = one-third x + 3
y = negative one-third x + 5
First, we need to find the slope:
m = (4 - 1) / (3 - (-6)) = 3/9 = 1/3
Next, we need to find the y-intercept (b) by substituting one of the points and the slope into the slope-intercept form:
1 = (1/3)(-6) + b
b = 3
Therefore, the equation of the line is y = (1/3)x + 3. Answer: y = one-third x + 3.
Which statements are correct steps in finding the linear equation of a line that passes through the points (−1, 7) and (2, 4) using the point-slope form method? Select all that apply.
y = x + 6
7 = –1(–1) + b
y – 4 = –1 (x – 2)
y – 7 = –1 (x – (–1))
y – 2 = x – 4
y = –x + 6
Some correct steps in finding the linear equation of a line that passes through the points (−1, 7) and (2, 4) using the point-slope form method are:
- Choose one of the given points to be the point (x1, y1) in the point-slope form.
- Find the slope of the line by using the slope formula with the two given points.
- Substitute the slope and the point (x1, y1) into the point-slope form and simplify.
So the correct statements are:
- 7 = –1(–1) + b
- y - 4 = -1(x - 2)
- y - 7 = -1(x - (-1))
incorrect
I'm sorry, could you please let me know which option is incorrect so that I can assist you better?
take a guess im helping you improve i cant give youu the wrong awnser you need to figur this out
Understood. Based on the given options, the statement that is incorrect as a step in finding the linear equation of a line that passes through the points (−1, 7) and (2, 4) using the point-slope form method is:
- y = x + 6
This is incorrect because it is the equation of a different line, not the one that passes through the given points.
nope
I apologize for my mistake. Could you please give me a hint on which option is incorrect?
7 = –1(–1) + b
I see. That is a correct step in finding the linear equation of the line. My apologies for the confusion earlier.
Use point-slope form, y - y1 = m(x - x1), to find the linear equation of a line that passes through the points
(2, -1) and (10, 7).
y =
x
+
First, let's find the slope m using the two points:
m = (7 - (-1))/(10 - 2) = 8/8 = 1
Now we can use one of the given points and the slope to write the equation in point-slope form:
y - (-1) = 1(x - 2)
Simplifying we have:
y + 1 = x - 2
Finally we can write the equation in slope-intercept form y = mx + b:
y = x - 3
Therefore, the linear equation of the line that passes through the points (2, -1) and (10, 7) is: y = x - 3.
Two different lines are shown below with two points given from each line. Use the slope-intercept form or point-slope form to find the equation of each line.
Line A
Points: (–5, –2), (–5, 7)
Line B
Points: (7, –5), (–2, –5)
Line A has
.
The equation of line A is
.
Line B has
.
The equation of line B is
.
For line A:
The x-coordinates of the two points are the same, which means that the line is vertical, and the slope is undefined.
The equation can be written in the form x = k, where k is the x-coordinate of the two points (i.e., x = -5).
Therefore, the equation of line A is x = -5.
For line B:
The y-coordinates of the two points are the same, which means that the line is horizontal, and the slope is zero.
Taking (7, -5) as the point, we know that:
y -(-5) = 0(x-7)
y + 5 = 0
y = -5
Therefore, the equation of line B is y = -5.
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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 Samuel and Helena can be correct, as the slope-intercept form and point-slope form can both be used to find the equation of a line given two points, depending on personal preference or the desired form of the equation.
Slope-intercept form of the equation of a line: y = mx + b
In the slope-intercept form of a line, we use the slope "m" and y-intercept "b" of the line to write the equation. To find the slope, we use the formula:
m = (y2 - y1) / (x2 - x1)
Once we have found "m," we can substitute "m" and the coordinates of one of the points to find "b" and write the equation in slope-intercept form.
Point-slope form of the equation of a line: y - y1 = m(x - x1)
In the point-slope form of a line, we use one of the given points (x1, y1), and the slope "m" of the line to find the equation. To find "m," we use the formula:
m = (y2 - y1) / (x2 - x1)
Then, we substitute the values into the point-slope form to get the equation.
Either form can be used to find the equation of a line given two points, and the choice between point-slope and slope-intercept forms usually depends on preference or the desired form of the equation.
Which can be the first step in finding the equation of the line that passes through the points (5, negative 4) and (negative 1, 8) in slope-intercept form?
Calculate StartFraction 8 minus (negative 4) Over negative 1 minus 5 EndFraction = StartFraction 12 Over negative 6 EndFraction = negative 2..
Calculate StartFraction negative 1 minus 5 Over 8 minus (negative 4) EndFraction = StartFraction negative 6 Over 12 EndFraction = negative one-half..
Find that the point at which the line intersects with the line y = 0 is (3, 0).
Find that the point at which the line intersects with the line x = y is (2, 2).
The first step in finding the equation of the line that passes through the points (5, -4) and (-1, 8) in slope-intercept form is to calculate the slope of the line.
We use the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) = (5, -4) and (x2, y2) = (-1, 8)
m = (8 - (-4)) / (-1 - 5) = 12/-6 = -2
Therefore, the slope of the line is -2.
One possible next step would be to find the y-intercept of the line "b" by substituting one of the given points and the slope into the slope-intercept form:
y = mx + b
Let's use the point (5, -4):
-4 = (-2)(5) + b
b = 6
Therefore, the equation of the line in slope-intercept form is:
y = -2x + 6.
What is the equation of the line passing through the points (Two-fifths, StartFraction 19 Over 20 EndFraction) and (one-third, StartFraction 11 Over 12 EndFraction) in slope-intercept form?
y = one-half x minus three-fourths
y = one-half x + three-fourths
y = three-fourths x minus one-half
y = three-fourths x + one-half
First, we need to find the slope m using the two points:
m = (11/12 - 19/20) / (1/3 - 2/5) = (11/60) / (1/15) = 11/4
Next, we need to find the y-intercept (b) by substituting one of the points and the slope into the slope-intercept form:
19/20 = (11/4)(2/5) + b
b = 3/4
Therefore, the equation of the line in slope-intercept form is:
y = (11/4)x + 3/4
Simplifying, we have:
y = 11/4 x + 3/4
So the correct option is y = 11/4 x + 3/4.
Which line has an equation of y = negative 5 x + 4 in slope-intercept form?
a line passing through the points (1, 9) and (3, 19)
a line passing through the points (2, –14) and (4, –24)
a line passing through the points (1, 1) and (3, 11)
a line passing through the points (2, –6) and (4, –16)
The given equation is y = -5x + 4, which is in slope-intercept form y = mx + b, where the slope (m) is -5 and the y-intercept (b) is 4.
To find which line has this equation, we can calculate the slope for each line, to see if it matches.
a) The slope for the line passing through (1, 9) and (3, 19) is:
m = (19 - 9)/(3 - 1) = 10/2 = 5
Therefore, this line does not have the same slope as the given equation.
b) The slope for the line passing through (2, -14) and (4, -24) is:
m = (-24 - (-14))/(4 - 2) = -10/2 = -5
Therefore, this line does have the same slope as the given equation.
c) The slope for the line passing through (1, 1) and (3, 11) is:
m = (11 - 1)/(3 - 1) = 10/2 = 5
Therefore, this line does not have the same slope as the given equation.
d) The slope for the line passing through (2, -6) and (4, -16) is:
m = (-16 - (-6))/(4 - 2) = -10/2 = -5
Therefore, this line does have the same slope as the given equation.
So, the lines that have the same slope as the given equation are options (b) and (d). However, only option (b) passes through the y-intercept 4, making it the line that has an equation of y = -5 x + 4 in slope-intercept form.
A)a line passing through the points (1, 9) and (3, 19)
B)a line passing through the points (2, –14) and (4, –24)
C)a line passing through the points (1, 1) and (3, 11)
D)a line passing through the points (2, –6) and (4, –16)
The line that has an equation of y = -5x + 4 in slope-intercept form is (B) a line passing through the points (2, -14) and (4, -24).
Lindsay used two points, (x 1, y 1) and (x 2, y 2), to find the equation of the line, y = mx + b, that passes through the points. First, she used the definition of slope and determined that the value of m is StartFraction y 2 minus y 1 Over x 2 minus x 1 EndFraction. Given this information, which expression must represent the value of b?
A)y 1 minus (StartFraction y 2 minus y 1 Over x 2 minus x 1 EndFraction) (x 1)
B)y 1 minus (StartFraction y 2 minus y 1 Over x 2 minus x 1 EndFraction) (x 2)
C)y 1 + (StartFraction y 2 minus y 1 Over x 2 minus x 1 EndFraction) (x 1)
D)y 1 + (StartFraction y 2 minus y 1 Over x 2 minus x 1 EndFraction) (x 2)
To determine which expression represents the value of b, we need to know the definition of y-intercept. The y-intercept is the point at which the line intersects the y-axis. That is, it is the value of y when x = 0.
In the slope-intercept form of the equation of a line, y = mx + b, "b" represents the y-intercept. So, to find the value of "b," we need to know the y-coordinate of any point on the line. Since the line passes through (x1, y1) and (x2, y2), we can use either point.
We know that the slope of the line is equal to:
m = (y2 - y1) / (x2 - x1)
Rearranging this equation, we get:
y2 - y1 = m(x2 - x1)
Substituting y1 and m in the slope-intercept form of the equation, we get:
y = mx + (y1 - mx1)
So, the correct expression representing the value of b is:
b = y1 - mx1
Substituting the given value of m, we get:
b = y1 - [(y2 - y1) / (x2 - x1)] * x1
Distributing the negative sign, we get:
b = y1 - (y2 - y1) / (x2 - x1) * x1
Simplifying, we have:
b = y1 - (y2 - y1) / (x2 - x1) * x1
Option (A) and (C) have a negative sign preceding the fraction and option (B) and (D) have a positive sign preceding the fraction. Since the slope is (y2 - y1) / (x2 - x1), which is the change in y over change in x, the y-intercept should be added since it is a y-value. Therefore, the correct option is (D) y1 + (y2 - y1) / (x2 - x1) * x1.