Question

How do you write the point slope form of the equation given \[(5,5)\] and slope \[3/5\]?

Answer 1

Hint: Use point-slope formula to write the equation of the line passing through the given point and having the slope of the given equation of line.
* Point-slope formula: Equation of line passing through the point \[({x_1},{y_1})\] and having slope ‘m’ is given by \[y - {y_1} = m(x - {x_1})\].

Complete step-by-step answer:
We are given a line having slope \[3/5\] and that the line passes through the point \[(5,5)\]
We know that the equation of line having slope m and passing through the point \[({x_1},{y_1})\] can be written using the point slope formula i.e. \[y - {y_1} = m(x - {x_1})\]
Comparing the values given in the question we get \[m = \dfrac{3}{5}\] and point \[({x_1},{y_1}) = (5,5)\]
Substitute the value of \[m = \dfrac{3}{5}\]and \[{x_1} = 5;{y_1} = 5\] in the point slope formula
\[ \Rightarrow y - 5 = \dfrac{3}{5}(x - 5)\]

\[\therefore \] The point slope form of the line passing through point \[(5,5)\] and having slope \[\dfrac{3}{5}\] is \[y - 5 = \dfrac{3}{5}(x - 5)\]

Note:
Many students make the mistake of solving the complete equation and write the final answer by bringing all terms to the left side of the equation which is not required here. We have to give the answer in point-slope form to which we can see and tell the value of the point and the slope.
Additional Information:
We can further solve the equation to bring it general form i.e. \[y = mx + c\]
Cross multiply the values from the denominator of the right hand side of the equation to the numerator of the left hand side of the equation.
\[ \Rightarrow 5 \times (y - 5) = 3 \times (x - 5)\]
Multiply the terms outside the bracket to the terms inside the brackets on both sides of the equation
\[ \Rightarrow 5 \times y - 5 \times 5 = 3 \times x - 3 \times 5\]
Calculate the products on both sides of the equation
\[ \Rightarrow 5y - 25 = 3x - 15\]
Bring all constant values to right hand side of the equation
\[ \Rightarrow 5y = 3x - 15 + 25\]
\[ \Rightarrow 5y = 3x + 10\]
Divide both sides by 5
\[ \Rightarrow y = \dfrac{3}{5}x + \dfrac{{10}}{5}\]
Cancel same factors from numerator and denominator on right hand side of the equation
\[ \Rightarrow y = \dfrac{3}{5}x + 2\]
So, the general equation of line passing through the point \[(5,5)\]and having slope \[\dfrac{3}{5}\]is \[y = \dfrac{3}{5}x + 2\].