Question

Find an equation of the line having the given slope and containing the given point. m = 6, (3,1)

Answer 1

To find the equation of a line with a slope of 6 that passes through the point (3,1), use the point-slope form and plug in the values to get y = 6x - 17.

Step-by-step explanation:

The question asks to find an equation of a line with a given slope and containing a specific point. The slope (m) is 6 and the line runs through point (3,1). One can use the point-slope form to write the equation of the line, which is:

y - y1 = m(x - x1)

Here, (x1, y1) is the given point (3,1), and m is the slope of 6. Plugging in these values gives:

y - 1 = 6(x - 3)

Next, distribute the slope through the parenthesis:

y - 1 = 6x - 18

Add 1 to both sides to solve for y:

y = 6x - 17

So, the equation of the line with slope 6 that passes through the point (3,1) is y = 6x - 17.

Answer 2

Main Answer:

The equation of the line with slope
`\(m = 6\)` and passing through the point
`\((3,1)\)` is
`\(y = 6x - 17\)`.

Step-by-step explanation:

To find the equation of a line, we can use the point-slope form:
`\(y - y_1 =` m
`(x - x_1)\),` where
`\(m\)` is the slope and
`\((x_1, y_1)\)` is a point on the line. Given \(m = 6\) and the point
`\((3,1)\)`, substitute these values into the formula. This yields
`\(y - 1 = 6(x - 3)\),` which can be simplified to
`\(y = 6x - 17\)`.

In this equation,
`\(6\)` represents the slope, indicating that for every unit increase in
`\(x\), \(y\)` increases by
`\(6\)`. The point
`\((3,1)\)` satisfies the equation, as plugging in
`\(x = 3\)` yields
`\(y = 1\)`. Therefore, the line passes through the given point.

Understanding how to manipulate the point-slope form allows us to determine the equation of a line with a specified slope and containing a given point. In this case, the slope
`\(m = 6\)` ensures the line's inclination, while the point
`\((3,1)\)` locks in a specific location on the line. The resulting equation,
`\(y = 6x - 17\)`, encapsulates both these aspects.