To find the equation of a line that passes through the point (6,3) and is perpendicular to the line represented by the equation 4x + 5y = 10, we need to follow a few steps involving some basic concepts of linear equations and slopes.
Step 1: Find the slope of the given line
The first step is to rewrite the equation of the line in slope-intercept form (y = mx + b), where ‘m’ represents the slope of the line.
Starting with the original equation:
4x + 5y = 10
We can solve for ‘y’:
- Subtract 4x from both sides:
- 5y = -4x + 10
- Now, divide every term by 5:
- y = - rac{4}{5}x + 2
From this, we can see that the slope (m) of the original line is -4/5.
Step 2: Find the slope of the perpendicular line
The slope of a line that is perpendicular to another line is the negative reciprocal of the original line’s slope. Therefore, to find the slope (mperpendicular) of the line we are interested in:
mperpendicular = - rac{1}{m} = - rac{1}{- rac{4}{5}} = rac{5}{4}
Step 3: Write the equation of the perpendicular line
Now that we have the slope of the perpendicular line, we can use the point-slope form of the equation of a line:
y – y1 = m(x – x1)
Where (x1, y1) is the point (6,3) and ‘m’ is the slope we just calculated, which is 5/4.
Substituting in the values:
- y – 3 = (5/4)(x – 6)
Now, let’s simplify this equation:
- y – 3 = (5/4)x – (30/4)
- y – 3 = (5/4)x – 7.5
- y = (5/4)x – 7.5 + 3
- y = (5/4)x – 4.5
Step 4: Final equation
The equation of the line that passes through the point (6,3) and is perpendicular to the line 4x + 5y = 10 is:
y = rac{5}{4}x – 4.5
This equation can also be written in standard form as needed, but this form clearly illustrates the slope and the y-intercept.