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Point Gradient and Two Point Formula>
Point Gradient and Two Point Formula 1Point Gradient and Two Point Formula 1
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Question 1 of 8
1. Question
Find the equation of a line with the gradient `m=2/5` and passing through the point `(-2,4)`.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)=(-2,4)`Gradient: `m=2/5``y-``y_1` `=` `m``(x-``x_1``)` Point-Gradient Formula `y-``4` `=` `(2/5)``(x-``(-2)``)` Plug in the values `y-4` `=` `2/5x + 4/5` `5y-20` `=` `2x + 4` Multiply both sides by `5` `5y-20` `-5y+20` `=` `2x + 4` `-5y+20` Add `-5y+20` to both sides `0` `=` `2x-5y+24` Simplify `2x-5y+24` `=` `0` `2x-5y+24=0` -
Question 2 of 8
2. Question
Find the equation of a line with the gradient `m=-1` and passing through the point `(4,-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)=(4,-3)`Gradient: `m=-1``y- color(royalblue)(y_1)` `=` `color(tomato)(m)(x- color(royalblue)(x_1))` Point Gradient formula `y- color(royalblue)(-3)` `=` `color(tomato)((-1))(x- color(royalblue)((4)))` Plug in the values `y+3` `=` `-x + 4` `-y-3` `x – 4` Divide both sides by `-1` `-y-3 color(crimson)(+y+3)` `=` `x – 4 color(crimson)(+y+3)` Add `y+3` to both sides `0` `=` `x + y -1` Simplify `x + y -1` `=` `0` `x + y -1=0` -
Question 3 of 8
3. Question
Find the equation of a line with the gradient `m=-2` and passing through the point `(3,5).`
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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)=(3,5)`Gradient: `m=-2``y- color(royalblue)(y_1)` `=` `color(tomato)(m)(x- color(royalblue)(x_1))` Point Gradient formula `y- color(royalblue)(5)` `=` `color(tomato)((-2))(x- color(royalblue)((3)))` Plug in the values `y-5` `=` `-2x + 6` `-y+5` `=` `2x – 6` Multiply both sides by `-1` `-y+5 color(crimson)(+y-5)` `=` `2x – 6 + color(crimson)(+y-5)` Add `+y-5` to both sides `0` `=` `2x+y-11` Simplify `2x+y-11` `=` `0` `2x+y-11=0` -
Question 4 of 8
4. Question
Find the equation of a line with the slope `m=0` and passing through the point `(4,-2).`
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Point Slope Form: `y- color(royalblue)(y_1)= color(tomato)(m)(x- color(royalblue)(x_1))`
- `color(tomato)(m)` is the slope of the line
- `(color(royalblue)(x_1,y_1))` is a point that lies on the line
Substitute the given slope and point into the formula.Point: `(x_1,y_1)=(4,-2)`Slope: `m=0``y- color(royalblue)(y_1)` `=` `color(tomato)(m)(x- color(royalblue)(x_1))` Point Slope formula `y- color(royalblue)(-2)` `=` `color(tomato)((0))(x- color(royalblue)((4)))` Plug in the values `y+2` `=` `0` `y+2=0` -
Question 5 of 8
5. Question
Find the equation of a line passing through the points `(0,2)` and `(2,-4)`.Hint
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Two Point Formula
$$\frac{y-\color{#9a00c7}{y_1}}{x-\color{#00880a}{x_1}}=\frac{\color{#9a00c7}{y_2}-\color{#9a00c7}{y_1}}{\color{#00880a}{x_2}-\color{#00880a}{x_1}}$$Solve for the gradient first using the given points.Point: `(``x_1`,`y_1``)=(``0`,`2``)`Point: `(``x_2`,`y_2``)=(``2`,`-4``)`$$\frac{y-\color{#9a00c7}{y_1}}{x-\color{#00880a}{x_1}}$$ `=` $$\frac{\color{#9a00c7}{y_2}-\color{#9a00c7}{y_1}}{\color{#00880a}{x_2}-\color{#00880a}{x_1}}$$ Two Point Formula $$\frac{y-\color{#9a00c7}{2}}{x-\color{#00880a}{0}}$$ `=` $$\frac{\color{#9a00c7}{-4}-\color{#9a00c7}{2}}{\color{#00880a}{2}-\color{#00880a}{0}}$$ Plug in the values `(y-2)/(x)` `=` `(-6)/2` `(y-2)/x` `=` `-3` `y-2` `=` `-3x` Multiply both sides by `x` `y-2` `+2` `=` `-3x` `+2` Add `2` to both sides `y` `=` `-3x+2` Simplify `y=-3x+2` -
Question 6 of 8
6. Question
Find the equation of a line passing through the points `(1,5)` and `(-4,-10)`.Hint
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Excellent!
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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`,`5``)`Point: `(``x_2`,`y_2``)=(``-4`,`-10``)``m` `=` $$\frac{\color{#9a00c7}{y_2}-\color{#9a00c7}{y_1}}{\color{#00880a}{x_2}-\color{#00880a}{x_1}}$$ Gradient Formula `=` $$\frac{\color{#9a00c7}{-10}-\color{#9a00c7}{5}}{\color{#00880a}{-4}-\color{#00880a}{1}}$$ Plug in the Values `=` `(-15)/(-5)` `m` `=` `3` Substitute the calculated gradient and one point into the Point-Gradient Formula.Point: `(x_1,y_1)=(1,5)`Gradient: `m= 3``y-``y_1` `=` `m``(x-``x_1``)` Point-Gradient Formula `y-``5` `=` `(3)``(x-``1``)` Plug in the values `y-5` `=` `3x-3` `y-5` `+5` `=` `3x-3` `+5` Add `5` to both sides `y` `=` `3x+2` Simplify `y=3x+2` -
Question 7 of 8
7. Question
Find the equation of a line passing through the points `(0,0)` and `(-4,3)`.
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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)(0), color(darkviolet)(0))`Point: `(color(forestgreen)(x_2),color(darkviolet)(y_2))=(color(forestgreen)(-4), 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)(0))/(x-color(forestgreen)(0))` `=` `(color(darkviolet)(3)-color(darkviolet)(0))/(color(forestgreen)(-4)-color(forestgreen)(0))` Plug in the values `y/x` `=` `3/(-4)` `y/x` `=` `3/(-4)` `y` `=` `(3x)/(-4)` Multiply both sides by `x` `y` `=` `(-3x)/4` Simplify `y=(-3x)/4` -
Question 8 of 8
8. Question
Find the equation of a line passing through the points `(0,-3)` and `(0,1)`.
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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)(-3))`Point: `(color(forestgreen)(x_2),color(darkviolet)(y_2))=(color(forestgreen)(0), color(darkviolet)(1))``(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)(-3))/(x-color(forestgreen)(0))` `=` `(color(darkviolet)(1)-color(darkviolet)(-3))/(color(forestgreen)(0)-color(forestgreen)(0))` Plug in the values `(y-3)/(x)` `=` `4/0` `(y-3)/x` `=` `4/0` Since there is a ‘0’ in the denominator, then the equation is unknown. undefined
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