Question

3) Find the equation of the line:

a) with a gradient of 2 and cutting the y-axis at 7
b) with a gradient of -2 and passing through the point (2;4)
c) passing through the points (2; 3) and (-1; 2)
d) parallel to the x-axis cutting the y-axis at 5

Answer 1

Explanation:

this is very much doing the exact same things as the previous question, just with a little bit different numbers.

remember, gradient = slope.

the slope is always the factor of x in the slope-intercept form

y = ax + b

our in the point-slope form

y - y1 = a(x - x1)

"a" is the slope, b is the y-intercept (the y- value when x = 0).

(x1, y1) is a point on the line.

the slope is the ratio (y coordinate change / x coordinate change) when going from one point on the line to another.

a)

y = 2x + 7

b)

y - 4 = -2(x - 2) = -2x + 4

y = -2x + 8

c)

going from (2, 3) to (-1, 2)

x changes by -3 (from 2 to -1)

y charges by -1 (from 3 to 2)

the slope is -1/-3 = 1/3

we use one of the points, e.g. (2, 3)

y - 3 = (1/3)×(x - 2) = x/3 - 2/3

y = x/3 - 2/3 + 3 = x/3 - 2/3 + 9/3 = x/3 + 7/3

d)

y = 5

this is a horizontal line (parallel to the x-axis) and represents every point on the grid, for which y = 5.

the slope is 0/x = 0, as y never changes at all.

the y- intercept is 5, of course.

Answer 2

`\textsf{a) \quad $y=2x+7$}`

`\textsf{b) \quad $y=-2x+8$}`

`\textsf{c) \quad $y=(1)/(3)x+(7)/(3)$}`

`\textsf{d) \quad $y=5$}`

Explanation:

Part (a)

Slope-intercept form of a linear equation:

`y=mx+b`

where:

• m is the slope.
• b is the y-intercept.

Given values:

• Slope = 2
• y-intercept = 7

Substitute the given values into the formula to create the equation of the line:

`\implies y=2x+7`

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Part (b)

Point-slope form of a linear equation:

`y-y_1=m(x-x_1)`

where:

• m is the slope.
• (x₁, y₁) is a point on the line.

Given:

• Slope = -2
• (x₁, y₁) = (2, 4)

Substitute the given values into the formula to create the equation of the line:

`\implies y-4=-2(x-2)`

`\implies y-4=-2x+4`

`\implies y=-2x+8`

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Part (c)

Slope formula:

`\textsf{slope}\:(m)=(y_2-y_1)/(x_2-x_1)`

where (x₁, y₁) and (x₂, y₂) are points on the line.

Given points:

• (x₁, y₁) = (2, 3)
• (x₂, y₂) = (-1, 2)

Substitute the points into the slope formula to calculate the slope of the line:

`\implies m=(2-3)/(-1-2)=(-1)/(-3)=(1)/(3)`

Substitute the found slope and one of the points into the point-slope formula to create the equation of the line:

`\implies y-y_1=m(x-x_1)`

`\implies y-3=(1)/(3)(x-2)`

`\implies y-3=(1)/(3)x-(2)/(3)`

`\implies y=(1)/(3)x+(7)/(3)`

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Part (d)

Slope-intercept form of a linear equation:

`y=mx+b`

where:

• m is the slope.
• b is the y-intercept.

If the line is parallel to the x-axis, its slope is zero.

If the line intersects the y-axis at y = 5, then its y-intercept is 5.

Therefore:

• m = 0
• b = 5

Substitute the given values into the formula to create the equation of the line:

`\implies y=0x + 5`

`\implies y=5`