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Point Gradient and Two Point Formula>
Point Gradient and Two Point Formula 2Point Gradient and Two Point Formula 2
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Question 1 of 7
1. Question
Find the equation of a line with the gradient `m=-1/2` and passing through the point `(5,-3)`.Hint
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Point Gradient Form: `y-``y_1``=``m``(x-``x_1``)`
- `m` is the gradient of the line
- `(x_1,y_1)` is a point that lies on the line
Substitute the given gradient and point into the formula.Point: `(x_1,y_1)=(5,-3)`Gradient: `m=-1/2``y-``y_1` `=` `m``(x-``x_1``)` Point-Gradient Formula `y-``(-3)` `=` `(-1/2)``(x-``5``)` Plug in the values `y+3` `=` `-1/2x+5/2` `2y+6` `=` `-x+5` Multiply both sides by `2` `2y+6` `+x-5` `=` `-x+5` `+x-5` Add `x-5` to both sides `x+2y+1` `=` `0` Simplify `x+2y+1=0` -
Question 2 of 7
2. Question
Find the equation of a line with the gradient `m=2` and passing through the point `(-5,-6).`
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Point Gradient Form: `y- color(royalblue)(y_1)= color(tomato)(m)(x- color(royalblue)(x_1))`
- `color(tomato)(m)` is the gradient of the line
- `(color(royalblue)(x_1,y_1))` is a point that lies on the line
Substitute the given gradient and point into the formula.Point: `(x_1,y_1)=(-5,-6)`Gradient: `m=2``y- color(royalblue)(y_1)` `=` `color(tomato)(m)(x- color(royalblue)(x_1))` Point Gradient formula `y- color(royalblue)(-6)` `=` `color(tomato)((2))(x- color(royalblue)((-5)))` Plug in the values `y+6` `=` `2x + 10` `y+6 color(crimson)(-y-6)` `=` `2x + 10 color(crimson)(-y-6)` Add `-y-6` to both sides `0` `=` `2x-y+4` Simplify `2x-y+4` `=` `0` `2x-y+4=0` -
Question 3 of 7
3. Question
Find the equation of a line with the gradient `m=-1/2` and passing through the point `(2,3)`.
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Point Gradient Form: `y- color(royalblue)(y_1)= color(tomato)(m)(x- color(royalblue)(x_1))`
- `color(tomato)(m)` is the gradient of the line
- `(color(royalblue)(x_1,y_1))` is a point that lies on the line
Substitute the given gradient and point into the formula.Point: `(x_1,y_1)=(2,3)`Gradient: `m=-1/2``y- color(royalblue)(y_1)` `=` `color(tomato)(m)(x- color(royalblue)(x_1))` Point Gradient formula `y- color(royalblue)(3)` `=` `color(tomato)((-1/2))(x- color(royalblue)((2)))` Plug in the values `y-3` `=` `-1/2x + 1` `2y-6` `=` `x – 2` Multiply both sides by `-2` `-2y+6 color(crimson)(+2y-6)` `=` `x – 2 color(crimson)(+2y-6)` Add `2y-6` to both sides `0` `=` `x+2y-8` Simplify `x+2y-8` `=` `0` `x+2y-8=0` -
Question 4 of 7
4. Question
Find the equation of a line passing through the points `(-1,-7)` and `(1,3)`.
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Two Point Formula
`(y-color(darkviolet)(y_1))/(x-color(forestgreen)(x_1))=(color(darkviolet)(y_2)-color(darkviolet)(y_1))/(color(forestgreen)(x_2)-color(forestgreen)(x_1))`Solve for the gradient first using the given points.Point: `(color(forestgreen)(x_1),color(darkviolet)(y_1))=(color(forestgreen)(-1), color(darkviolet)(-7))`Point: `(color(forestgreen)(x_2),color(darkviolet)(y_2))=(color(forestgreen)(1), color(darkviolet)(3))``(y-color(darkviolet)(y_1))/(x-color(forestgreen)(x_1))` `=` `(color(darkviolet)(y_2)-color(darkviolet)(y_1))/(color(forestgreen)(x_2)-color(forestgreen)(x_1))` Two Point Formula `(y-color(darkviolet)(-7))/(x-color(forestgreen)(-1))` `=` `(color(darkviolet)(3)-color(darkviolet)(-7))/(color(forestgreen)(1)-color(forestgreen)(-1))` Plug in the values `(y+7)/(x+1)` `=` `(10)/2` `(y+7)/(x+1)` `=` `5` `y+7` `=` `5x+5` Multiply both sides by `x+1` `y+7 color(crimson)(-2)` `=` `5x+5 color(crimson)(-2)` Add `-2` to both sides `y` `=` `5x+3` Simplify `y=5x+3` -
Question 5 of 7
5. Question
Find the equation of a line passing through the points `(0,2)` and `(1,-2)`.
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Two Point Formula
`(y-color(darkviolet)(y_1))/(x-color(forestgreen)(x_1))=(color(darkviolet)(y_2)-color(darkviolet)(y_1))/(color(forestgreen)(x_2)-color(forestgreen)(x_1))`Solve for the gradient first using the given points.Point: `(color(forestgreen)(x_1),color(darkviolet)(y_1))=(color(forestgreen)(0), color(darkviolet)(2))`Point: `(color(forestgreen)(x_2),color(darkviolet)(y_2))=(color(forestgreen)(1), color(darkviolet)(-2))``(y-color(darkviolet)(y_1))/(x-color(forestgreen)(x_1))` `=` `(color(darkviolet)(y_2)-color(darkviolet)(y_1))/(color(forestgreen)(x_2)-color(forestgreen)(x_1))` Two Point Formula `(y-color(darkviolet)(2))/(x-color(forestgreen)(0))` `=` `(color(darkviolet)(-2)-color(darkviolet)(2))/(color(forestgreen)(1)-color(forestgreen)(0))` Plug in the values `(y-2)/(x)` `=` `(-4)/1` `(y-2)/x` `=` `-4` `y-2` `=` `-4x` Multiply both sides by `x` `y-2 color(crimson)(+2)` `=` `-4x color(crimson)(+2)` Add `2` to both sides `y` `=` `-4x+2` Simplify `y=-4x+2` -
Question 6 of 7
6. Question
Find the equation of a line passing through the points `(-6,8)` and `(4,8)`.
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Two Point Formula
`(y-color(darkviolet)(y_1))/(x-color(forestgreen)(x_1))=(color(darkviolet)(y_2)-color(darkviolet)(y_1))/(color(forestgreen)(x_2)-color(forestgreen)(x_1))`Solve for the gradient first using the given points.Point: `(color(forestgreen)(x_1),color(darkviolet)(y_1))=(color(forestgreen)(-6), color(darkviolet)(8))`Point: `(color(forestgreen)(x_2),color(darkviolet)(y_2))=(color(forestgreen)(4), color(darkviolet)(8))``(y-color(darkviolet)(y_1))/(x-color(forestgreen)(x_1))` `=` `(color(darkviolet)(y_2)-color(darkviolet)(y_1))/(color(forestgreen)(x_2)-color(forestgreen)(x_1))` Two Point Formula `(y-color(darkviolet)(8))/(x-color(forestgreen)(-6))` `=` `(color(darkviolet)(8)-color(darkviolet)(8))/(color(forestgreen)(4)-color(forestgreen)(-6))` Plug in the values `(y-8)/(x+6)` `=` `0/(4+6)` `(y-8)/(x+6)` `=` `0` `y-8` `=` `0` `y-8 color(crimson)(+8)` `=` `0 color(crimson)(+8)` Add `8` to both sides `y` `=` `8` Simplify `y=8` -
Question 7 of 7
7. Question
Find the equation of a line passing through the points `(1,8)` and `(-2,2)`.Hint
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Point Gradient Form: `y-``y_1``=``m``(x-``x_1``)`
- `m` is the gradient of the line
- `(x_1,y_1)` is a point that lies on the line
Gradient Formula
$$m=\frac{\color{#9a00c7}{y_2}-\color{#9a00c7}{y_1}}{\color{#00880a}{x_2}-\color{#00880a}{x_1}}$$Solve for the gradient first using the given points.Point: `(``x_1`,`y_1``)=(``1`,`8``)`Point: `(``x_2`,`y_2``)=(``-2`,`2``)``m` `=` $$\frac{\color{#9a00c7}{y_2}-\color{#9a00c7}{y_1}}{\color{#00880a}{x_2}-\color{#00880a}{x_1}}$$ Gradient Formula `=` $$\frac{\color{#9a00c7}{2}-\color{#9a00c7}{8}}{\color{#00880a}{-2}-\color{#00880a}{1}}$$ Plug in the Values `=` `(-6)/(-3)` `m` `=` `2` Substitute the calculated gradient and one point into the Point-Gradient Formula.Point: `(x_1,y_1)=(1,8)`Gradient: `m= 2``y-``y_1` `=` `m``(x-``x_1``)` Point-Gradient Formula `y-``8` `=` `(2)``(x-``1``)` Plug in the values `y-8` `=` `2x-2` `y-8` `+8` `=` `2x-2` `+8` Add `8` to both sides `y` `=` `2x+6` Simplify `y=2x+6`
Quizzes
- Distance Between Two Points 1
- Distance Between Two Points 2
- Distance Between Two Points 3
- Midpoint of a Line 1
- Midpoint of a Line 2
- Midpoint of a Line 3
- Gradient of a Line 1
- Gradient of a Line 2
- Gradient Intercept Form: Graph an Equation 1
- Gradient Intercept Form: Graph an Equation 2
- Gradient Intercept Form: Write an Equation 1
- Determine if a Point Lies on a Line
- Graph Linear Inequalities 1
- Graph Linear Inequalities 2
- Convert Between General Form and Gradient Intercept Form 1
- Convert Between General Form and Gradient Intercept Form 2
- Point Gradient and Two Point Formula 1
- Point Gradient and Two Point Formula 2
- Parallel Lines 1
- Parallel Lines 2
- Perpendicular Lines 1
- Perpendicular Lines 2