Question

Write an equation of the line that passes through (1 3) and has a slope of 5/4

Answer 1

`(\stackrel{x_1}{1}~,~\stackrel{y_1}{3})\hspace{10em} \stackrel{slope}{m} ~=~ \cfrac{5}{4} \\\\\\ \begin{array}c \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{3}=\stackrel{m}{ \cfrac{5}{4}}(x-\stackrel{x_1}{1}) \\\\\\ y-3=\cfrac{5}{4}x-\cfrac{5}{4}\implies y=\cfrac{5}{4}x-\cfrac{5}{4}+3\implies {\Large \begin{array}{llll} y=\cfrac{5}{4}x+\cfrac{7}{4} \end{array}}`

Answer 2

The equation of the line passing through the point (1, 3) with a slope of 5/4 is
`\( y - 3 = (5)/(4)(x - 1) \).`

To find the equation of a line, we can use the point-slope form, which is
`\( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \)` is a point on the line, and ( m ) is the slope. Given the point (1, 3) and a slope of
`\( (5)/(4) \)`, we substitute these values into the point-slope form, resulting in
`\( y - 3 = (5)/(4)(x - 1) \).`

This equation can be further simplified to the slope-intercept form, ( y = mx + b ), where ( m ) is the slope and ( b ) is the y-intercept. Simplifying the equation, we get
`\( y = (5)/(4)x - (5)/(4) + 3 \)`, which can be written as
`\( y = (5)/(4)x + (7)/(4) \).`

In summary, the equation
`\( y - 3 = (5)/(4)(x - 1) \)` represents a line with a slope of
`\( (5)/(4) \)` passing through the point (1, 3). Understanding how to derive the equation of a line from a given point and slope is fundamental in algebra and geometry, providing a powerful tool for describing linear relationships in mathematical models and real-world scenarios.