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Point gradient formula is an equation used to determine the slope of any given line. It is used to determine the slope of a line, and the point through which it travels. It usually gives an equation to determine the nature of a straight line. This straight line should be inclined at a particular angle on the X-axis and must pass through one of the points. To determine an equation for a straight line using the point gradient formula, the slope of the line must be known, the slope is represented by m.
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Read More: Introduction to Three-Dimensional Geometry
Key Takeaways: Point slope form, Point slope gradient formula, Equation of Straight Line.
Point Slope Form
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Point-slope formula is one of the four different formulas which are used to determine the equation for a straight line. This straight line should be inclined at a particular angle on the X-axis and must pass through one of the points. To determine an equation for a straight line using the point gradient formula, the slope of the line must be known, the slope is represented by m. The other formulas used to determine the equation of a line are:
- General formula
- Gradient intercept formula
- Intercept formula
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Point Slope Gradient Formula
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Before determining the point gradient formula, we should possess a line. All the lines, except the lines which are parallel to the X and Y axis on the Cartesian plane, lie between two points and pass through some points. For example, a line, l, passes through the points (a,0) and (b,0). Then a is the incept lying on X-axis and b is the intercept lying on the y axis.
Straight Line Equation
The point gradient formula for a point P (x,y) on a line l, having slope m, passing through points (x1, y1), and lying at an angle θ from X-axis is given by:
m = (y – y1)(x – x1)
m(x – x1) = y – y1
Here,
x, and y = known coordinates of the point P
x1 and y1 = random points on the line which can be kept as variables.
m = slope of the line
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Derivation of Point Gradient Formula
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To derive the formula for point-slope gradient, we need to make a few assumptions, they are, we need to derive the point gradient formula for a point P (x1,y1) on a line l, having slope m, passing through points (x, y), and lying at an angle θ from X-axis is given by,
Method 1:
We know that the slope, m of a line is:
Slope, m = tan θ
Tan θ = perpendicular / base
Perpendicular = y-y1
Base = x-x1
m = tan θ = = (y-y1)(x-x1)
m(x-x1) = y-y1
Method 2:
We know that the slope, m of a line is:
Slope, m = tan θ
m = (Difference in y-coordinates)/(Difference in x-coordinates)
m = (y-y1)(x-x1)
Multiplying both the sides with x-x1, we get:
m(x-x1) = y-y1
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How to Use Point Gradient Formula
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To determine the point gradient formula for a given straight line, the following steps can be followed:
Step 1: Determine the angle at which the line lies from the X-axis.
Step 2: Determine the slope of the line using the angle between the line and the X-axis
Step 3: Determine the variable coordinates and the coordinates of the point whose point gradient has to be calculated.
Step 4: Use the point gradient formula, and derive the equation.
Other formulas used to determine the equation of a straight line:
General Formula:
For a line, l, passes through the points (a,0) and (b,0). Then a is the incept lying on X-axis and b is the intercept lying on the y axis.
ax + by + c = 0
where a and b both are not 0
Gradient Intercept Formula
This is derived from the point gradient formula.
We know that m(x – x1) = y – y1,
if (x1, y1) = (0,c).
Then the equation changes to:
m(x) = y – C
This is the gradient intercept formula.
Two-Point Formula
For a line passing through two points having coordinates (x1, y1) and (y1, y2), the slope is given as
m(x2-x1) = y2-y1
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Things to Remember
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- There are four different formulas used to determine an equation of a straight line, general formula, point gradient formula, gradient intercept formula, and two-point formula.
- The general formula is ax+by+c=0, where a and b both are not 0
- The point gradient formula is, m(x-x1) = y-y1
- The gradient intercept formula is m(x) = y-C
- The two-point formula is, m(x2-x1) = y2-y1
- In the case of lines parallel to the x or y-axis, the point gradient formula cannot be calculated.
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Sample Questions
Ques: Determine the slope, that is m, for the equation of the line going through (1,3) and (5,1). (2 Marks)
Answer: Here two-point gradient formula can be used,
X1 = 1, Y1 = 3, X2 = 5, Y2 = 1,
m(x2-x1) = y2-y1
m = 5-1/1- 3
m = 4/-2
m = -2
Ques: Determine the slope for the equation of the straight line passing through points (−3;2) and (5;8). (2 Marks)
Answer: Here two-point gradient formula can be used,
X1 = -3, Y1 = 2, X2 = 5, Y2 = 8,
m(x2-x1) = y2-y1
m = 8-2/5-(-3)
m = 6/8
m = 3/4
Ques: Determine the equation for a line that passes through a point P having coordinates (2, -3) and having a slope (-1/2). (3 Marks)
Answer: The coordinates of the point P (2,-3)
Slope m = -1/2
To determine the point gradient, use the formula:
m(x-x1) = y-y1
-1/2[x-2] = y-(-3)
-x/2 + 1 = y +3
Now we need to subtract three from both the sides, we get,
Y = -x/2 – 2
Thus, the equation is Y = -x/2 – 2
Ques: Determine the equation for a line that passes through a point P having coordinates (-1, -5) and having a slope (4). (3 Marks)
Answer: The coordinates of the point P (-1,-5)
Slope m = 4
To determine the point gradient, use the formula:
m(x-x1) = y-y1
4[x-(-1)] = y-(-5)
4x + 4 = y +5
Now we need to subtract five from both the sides, we get,
Y = 4x +4 – 5
Thus, the equation is Y = 4x-1
Ques: Determine the equation of the straight line with gradient m=−13 and passing through the point (−1;1). (3 Marks)
Answer: The coordinates of the point P (-1,1)
Slope m = -13
To determine the point gradient, use the formula:
m(x-x1) = y-y1
-13[x-(-1)] = y-(1)
-13x + (-13) = y -1
Now we need to add one to both the sides, we get,
Y = -13x +1 – 13
Thus, the equation is Y = -13x + 12
Ques: Determine the equation of the straight line passing through points (−3;2) and slope 3/4 (3 Marks)
Answer: The coordinates of the point P (-3,2)
Slope m = 3/4
To determine the point gradient, use the formula:
m(x-x1) = y-y1
3/4[x-(-3)] = y-(2)
3/4x + 3 = y -2
Now we need to add 2 to both the sides, we get,
Y = 3/4x +5
Thus, the equation is Y = 3/4x +5
Ques: Determine the equation of the straight line passing through points (−3;2) and (5;8). (5 Marks)
Answer: Here two-point gradient formula can be used,
X1 = -3, Y1 = 2, X2 = 5, Y2 = 8,
m(x2-x1) = y2-y1
m = 8-2/5-(-3)
m = 6/8
m = 3/4
Now use the value of m in the formula below:
m(x-x1) = y-y1
3/4[x-(-3)] = y-(2)
3/4x + 3 = y -2
Now we need to add 2 to both the sides, we get,
Y = 3/4x +5
Thus, the equation is Y = 3/4x +5
Ques: Determine the equation of the straight line passing through points going through (1,3) and (5,1). (5 Marks)
Answer: Here two-point gradient formula can be used,
X1 = 1, Y1 = 3, X2 = 5, Y2 = 1,
m(x2-x1) = y2-y1
m = 5-1/1- 3
m = 4/-2
m = -2
Now use the value of m in the formula below:
m(x-x1) = y-y1
-2[x-(1)] = y-(3)
-2x + 2 = y -3
Now we need to add three to both the sides, we get,
Y = -2x +5
Thus, the equation is Y = -2x +5
Ques: Find the equation of the line through (3,4) and (−2, −3). (5 Marks)
Answer: Here two-point gradient formula can be used,
X1 = 3, Y1 = 4, X2 = -2, Y2 = -3,
m(x2-x1) = y2-y1
m = -2-3/-3- 4
m = -5/-7
m = 5/7
Now use the value of m in the formula below:
m(x-x1) = y-y1
5/7[x-(3)] = y-(4)
5/7x -3 = y -4
Now we need to add four to both the sides, we get,
Y = 5/7x +1
Thus, the equation is Y = 5/7x +1
Ques: Find the equation of the line through (4,4) and (−2, −3). (5 Marks)
Answer: Here two-point gradient formula can be used,
X1 = 4, Y1 = 4, X2 = -2, Y2 = -3,
m(x2-x1) = y2-y1
m = -2-4/-3- 4
m = -6/-7
m = 6/7
Now use the value of m in the formula below:
m (x – x1) = y – y1
5/7[x - (4)] = y – (4)
5/7x – 4 = y – 4
Now we need to add four to both the sides, we get,
Y = 5/7x
Thus, the equation is Y = 5/7
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