Write the standard form of the line that passes through the given points. Include your work in your final answer. Type your answer in the box provided or use the upload option to submit your solution.
(3, 1) and (-2, 3)
Write the slope-intercept form of the line that passes through the point (1, 0) and is parallel to x - y = 7. Type your answer in the box provided or use the upload option to submit your solution.
To find the equation of the line passing through two given points, we can use the point-slope form of a linear equation:
y - y1 = m(x - x1),
where (x1, y1) are the coordinates of one of the points, and m is the slope of the line.
First, let's find the slope (m) using the formula:
m = (y2 - y1) / (x2 - x1).
Given the points (3, 1) and (-2, 3), we can substitute these values into the formula:
m = (3 - 1) / (-2 - 3)
= 2 / (-5).
Now we have the slope. Let's choose one of the points, say (3, 1), and substitute its coordinates and the slope into the point-slope form equation:
y - 1 = (2 / -5) * (x - 3).
To remove the fraction, we can multiply both sides of the equation by -5:
-5y + 5 = 2x - 6.
Rearranging the equation, we get:
2x + 5y = 11.
Therefore, the standard form of the line passing through the points (3, 1) and (-2, 3) is 2x + 5y = 11.
To find the equation of a line in standard form that passes through two points, we can use the point-slope form and then convert it to standard form.
The point-slope form of a linear equation is y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope.
Given the points (3, 1) and (-2, 3), we can choose one of them as our reference point. Let's use (3, 1).
First, we need to find the slope (m) of the line. The slope is calculated using the formula:
m = (y₂ - y₁) / (x₂ - x₁)
Using the coordinates (3, 1) as our reference point:
m = (3 - 1) / (-2 - 3)
m = 2 / -5
Now that we have the slope, let's substitute the values into the point-slope form equation:
y - 1 = (2 / -5)(x - 3)
To convert this to standard form, we want the equation to be in the form Ax + By = C, where A, B, and C are integers.
Multiplying both sides of the equation by -5 to eliminate the fraction:
-5(y - 1) = 2(x - 3)
-5y + 5 = 2x - 6
2x + 5y = 11
Therefore, the standard form of the line that passes through the points (3, 1) and (-2, 3) is 2x + 5y = 11.
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