Question

The straight line that passes through the points (4, v) and (8, 1) has

gradient 5/7.
What is the value of v?
Give your answer as an integer or as a fraction in its simplest form.

Answer 1

If the gradient of a straight line that passes through the points (4, v) and (8, 1) is 5/7, then the value of v is:

`\boxed{v=-(13)/(7)}`

Explanation:

We can use the slope formula to find the value of v if a straight line passes through the point (4, v) and (8, 1).

`\boxed{\begin{minipage}{8cm}\underline{Slope Formula}\\\\Slope $(m)=(y_2-y_1)/(x_2-x_1)$\\\\where $(x_1,y_1)$ and $(x_2,y_2)$ are two points on the line.\\\end{minipage}}`

Define the points:

• Let (x₁, y₁) = (8, 1)
• Let (x₂, y₂) = (4, v)

Given that the gradient (slope) is 5/7, substitute the points into the formula and equate it to 5/7:

`\implies (v-1)/(4-8)=(5)/(7)`

Solve the equation for v.

Simplify the denominator:

`\implies (v-1)/(-4)=(5)/(7)`

Multiply both sides of the equation by -4:

`\implies (v-1)/(-4)\cdot(-4)=(5)/(7) \cdot(-4)`

`\implies v-1=-(20)/(7)`

Add 1 to both sides of the equation:

`\implies v-1+1=-(20)/(7)+1`

`\implies v=-(20)/(7)+1`

Convert +1 into a fraction with a denominator of 7:

`\implies v=-(20)/(7)+(7)/(7)`

`\textsf{Apply the fraction rule} \quad (a)/(c)-(b)/(c)=(a-b)/(c):`

`\implies v=(-20+7)/(7)`

`\implies v=-(13)/(7)`

Therefore, the value of v is:

`\boxed{v=-(13)/(7)}`