I have an assignment that asks me to write an equation in slope-intercept, point-slope, or standard form for the information given and to explain why the chosen form would be best. Below is the information given.
1. passing through (-1,4) and (-5,2)
2. that has slope 2 and y intercept 4
3. that has an x-intercept of 6 and a y-intercept of 3
4. that passes through (1,2) and has a slope of -5/3
I chose point-slope for the first one, slope-intercept for the second, slope-intercept for the third, and point-slope for the fourth. Do you think my answers are good, or would you choose different forms for some of them?
Your choices for the equation forms are mostly correct, but there are a couple of adjustments that can be made. Let's analyze each scenario and determine the most appropriate equation form.
1. Passing through (-1,4) and (-5,2):
For this situation, you correctly selected the point-slope form. The point-slope form of a linear equation is given by y - y1 = m(x - x1), where (x1, y1) are the coordinates of a point on the line, and m is the slope. In this case, you can choose (-1,4) as the point, and the slope can be calculated from the two points given:
m = (y2 - y1) / (x2 - x1)
= (2 - 4) / (-5 - (-1))
= -2 / (-5 + 1)
= -2 / -4
= 1/2
So, the equation in point-slope form is: y - 4 = (1/2)(x - (-1)).
2. Slope 2 and y-intercept 4:
Here, you correctly chose slope-intercept form. The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept. Given that the slope is 2 and the y-intercept is 4, the equation in slope-intercept form is: y = 2x + 4.
3. X-intercept of 6 and y-intercept of 3:
You initially chose slope-intercept form, which is incorrect for this scenario. Instead, we should use standard form, which is written as Ax + By = C. To determine the equation, we need to find the values of A, B, and C.
The x-intercept represents the point where the line intersects the x-axis, which occurs when y = 0. Therefore, the x-intercept is (6, 0).
The y-intercept represents the point where the line intersects the y-axis, which occurs when x = 0. Therefore, the y-intercept is (0, 3).
We can then use the two points to find the equation. Using the slope formula: m = (y2 - y1) / (x2 - x1), and plugging in the values, we have:
m = (0 - 3) / (6 - 0)
= -3 / 6
= -1/2
Now, we can substitute one of the points and the slope into the point-slope form and rearrange it into the standard form:
y - y1 = m(x - x1)
y - 3 = (-1/2)(x - 0)
y - 3 = (-1/2)x
Multiply both sides by 2 to eliminate the fraction:
2y - 6 = -x
To make the coefficient of x positive, we can multiply both sides by -1:
-x + 2y - 6 = 0
Therefore, the equation in standard form for the given information is: -x + 2y - 6 = 0.
4. Passing through (1,2) and a slope of -5/3:
You initially chose point-slope form, which is incorrect for this scenario. The most appropriate equation form here is slope-intercept form. Given that the slope is -5/3 and the point (1,2), the equation can be written as:
y - y1 = m(x - x1)
y - 2 = (-5/3)(x - 1)
Now, let's simplify it and convert it into slope-intercept form:
y - 2 = (-5/3)x + 5/3
y = (-5/3)x + 5/3 + 2
y = (-5/3)x + 5/3 + 6/3
y = (-5/3)x + 11/3
To summarize, the appropriate equation forms for each scenario are:
1. Point-slope form: y - 4 = (1/2)(x - (-1))
2. Slope-intercept form: y = 2x + 4
3. Standard form: -x + 2y - 6 = 0
4. Slope-intercept form: y = (-5/3)x + 11/3
Your choices are generally correct, except for the third scenario where the standard form is more appropriate.
Your choices for the equation forms are mostly correct. Let's go through each question and determine the best equation form for the given information:
1. Passing through (-1,4) and (-5,2):
To find the equation of a line given two points, we can use the point-slope form which is:
y - y1 = m(x - x1)
where (x1, y1) represents one of the given points and m is the slope of the line.
So, for the first question, the equation would be:
y - 4 = m(x - (-1))
You can substitute any of the given points into the equation to find the value of m. Let's use (-1,4):
4 - 4 = m(-1 - (-1))
0 = 2m (you may have gotten a different equation for m here, but that's okay)
Now, we can rewrite the equation as:
y - 4 = 2(x + 1)
2. Slope of 2 and y-intercept of 4:
The slope-intercept form of an equation is y = mx + b, where m is the slope and b is the y-intercept.
For the second question, the equation would be:
y = 2x + 4
3. X-intercept of 6 and y-intercept of 3:
The slope-intercept form is still the best choice for this question since the x-intercept and y-intercept values are given explicitly.
For the third question, we know that the point (x, 0) lies on the x-axis, so it will have a y-value of 0. Therefore, the equation becomes:
0 = mx + b
We are given the x-intercept (6, 0), so we can substitute these values into the equation to find the value of m:
0 = 6m + b
We are also given the y-intercept (0, 3), so we substitute this point as well:
3 = 0 + b
Now that we have the values of b and m, we can rewrite the equation as:
0 = 6x + 3
4. Passes through (1,2) and has a slope of -5/3:
Similar to the first question, we use point-slope form here.
For the fourth question, the equation would be:
y - 2 = m(x - 1)
To determine the value of m, we substitute the given slope:
y - 2 = (-5/3)(x - 1)
In conclusion, your choice of equation forms is mostly correct. However, for the third question, you mistakenly chose slope-intercept form instead of standard form. Therefore, the correct equation forms are:
1. Point-slope form: y - 4 = 2(x + 1)
2. Slope-intercept form: y = 2x + 4
3. Standard form: 0 = 6x + 3
4. Point-slope form: y - 2 = (-5/3)(x - 1)
Keep in mind that there may be multiple valid equation forms for each set of information, but I have provided the most commonly used forms for simplicity.
actually, any method should work for any of those conditions,
in #3, I would use the intercept-intercept form
if the x-intecept is 'a' and the y-intercept is 'b', then the equation can be written simply as
x/a + y/b = 1
so x/6 + y/3 = 1
times 6
x + 2y = 6 , all done!
If I know the slope and a point, I use a method that given me the equation in about 3 lines
- based on the fact that for
Ax + By = C, the slope is -A/B
if you give me the slope I can reverse that and start with the completed left side of the equation,
e.g.
given slope as -5/3 and the point (1,2) , your #4
the equation has to be
5x + 3y = C , but (1,2) lies on it, so
5(1) + 3(2) = C
C = 11
equation: 5x + 3y = 11
advantage, no fractions!