Calculating the Equation of a Line:
To calculate the equation of a line, you need to know the possible formats that the equation of a line can be written in:
1. y = mx + c :
- "The equation of the line with gradient m and y-intercept c"
- In this format, it is given that y = y-coordinate, m = gradient, x = x-coordinate and c = y-intercept.
- From this you can then calculate the x-intercept, and work out the values of x or y when one is known, and m and c are given.
2. y - y1 = m(x - x1) :
- "The equation of the line with gradient m which passes through the point (x1, y1)"
- In this format, it is given that y = y-coordinate, m = gradient, x = x-coordinate and c = y-intercept as normal, but also that (x1, y1) are then substituted into the formula.
3. ax + by + c = 0 :
- "The equation of the line where a, b and c are integers and is equal to zero"
- Equations 1 and 2 can be rearranged to take on this format.
Example:
1. Find the equation of the line parallel to 3x + 4y - 5 = 0 which passes through the point (5, -7).
Find the gradient (m) :
3x + 4y - 5 = 0 (-3x+5)
4y = -3x + 5 (/4)
y = -3/4x + 5/4
Therefore: m1 = -3/4 = m2
Use y - y1 = m(x - x1) :
y- -7 = -3/4(x-5)
y+7 = -3/4(x-5)
a) Give your answer in the form y = mx + c
y+7 = -3/4(x-5) (-7)
y = -3/4(x-5) - 7
y = -3/4x + 15/4 -7
y = -3/4x + 15/4 -28/4
y = -3/4x -13/4
b) Give your answer in the form ax + by + c = 0
y+7 = -3/4(x-5) (x4)
4y + 28 = -3(x-5)
4y + 28 = -3x + 15 (+3x-15)
4y + 3x +13 = 0
3x + 4y + 13 = 0 or -3x - 4y -13 = 0
Find the gradient (m) :
3x + 4y - 5 = 0 (-3x+5)
4y = -3x + 5 (/4)
y = -3/4x + 5/4
Therefore: m1 = -3/4 = m2
Use y - y1 = m(x - x1) :
y- -7 = -3/4(x-5)
y+7 = -3/4(x-5)
a) Give your answer in the form y = mx + c
y+7 = -3/4(x-5) (-7)
y = -3/4(x-5) - 7
y = -3/4x + 15/4 -7
y = -3/4x + 15/4 -28/4
y = -3/4x -13/4
b) Give your answer in the form ax + by + c = 0
y+7 = -3/4(x-5) (x4)
4y + 28 = -3(x-5)
4y + 28 = -3x + 15 (+3x-15)
4y + 3x +13 = 0
3x + 4y + 13 = 0 or -3x - 4y -13 = 0