Answer:
To find the equation of a line passing through two given points, we can use the point-slope form, slope-intercept form, and general form of a linear equation. Let's calculate the equation using these three forms.
1. 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 of the line.
Given points: (5, 5) and (7, 8)
Let's calculate the slope (m) first:
m = (y2 - y1) / (x2 - x1)
= (8 - 5) / (7 - 5)
= 3 / 2
Now, let's choose one of the points, say (5, 5), and substitute its coordinates into the point-slope form:
y - 5 = (3/2)(x - 5)
Simplifying this equation gives us the point-slope form:
y - 5 = (3/2)x - 15/2
y = (3/2)x - 15/2 + 10/2
y = (3/2)x - 5/2
Therefore, the equation of the line in point-slope form is y = (3/2)x - 5/2.
2. Slope-Intercept Form:
The slope-intercept form of a linear equation is given by y = mx + b, where m is the slope and b is the y-intercept.
From our previous calculations, we already know that the slope (m) is equal to 3/2. To find the y-intercept (b), we can substitute one of the given points into the equation and solve for b.
Using the point (5, 5):
5 = (3/2)(5) + b
5 = 15/2 + b
10/2 - 15/2 = b
-5/2 = b
Now we can write the equation of the line in slope-intercept form:
y = (3/2)x - 5/2
Therefore, the equation of the line in slope-intercept form is y = (3/2)x - 5/2.
3. General Form:
The general form of a linear equation is given by Ax + By = C, where A, B, and C are constants.
To convert the equation from slope-intercept form to general form, we multiply both sides of the equation by 2 to eliminate fractions:
2y = 3x - 5
Rearranging the terms gives us:
-3x + 2y = -5
Therefore, the equation of the line in general form is -3x + 2y = -5.
In summary, the equation of the line passing through the points (5, 5) and (7, 8) can be written as:
- Point-Slope Form: y = (3/2)x - 5/2
- Slope-Intercept Form: y = (3/2)x - 5/2
- General Form: -3x + 2y = -5
Explanation: