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There is a line that includes the point (6, 4) and has a slope of 5. What is its equation in point-slope form?



Use the specified point in your equation. Write your answer using integers, proper fractions, and improper fractions. Simplify all fractions.

Verified Answer :

  • Identify the point-slope formula: \(y - y_1 = m(x - x_1)\).
  • Substitute the given point \((6, 4)\) and the slope \(5\) into the formula.
  • The equation in point-slope form is \(\boxed{y - 4 = 5(x - 6)}\).

Explanation

  1. Analyzing the Problem and Recalling the Formula

To find the equation of a line in point-slope form, we use the formula \(y - y_1 = m(x - x_1)\), where \(m\) is the slope of the line and \((x_1, y_1)\) is a known point on the line. In this problem, we are given the point \((x_1, y_1) = (6, 4)\) and the slope \(m = 5\).

  1. Substituting the Values into the Formula

Substituting the given values \(m = 5\), \(x_1 = 6\), and \(y_1 = 4\) into the point-slope formula \(y - y_1 = m(x - x_1)\), we get:
\[y - 4 = 5(x - 6)\]
This equation is now in the required point-slope form.

  1. Final Conclusion

The final equation of the line passing through the point \((6, 4)\) with a slope of \(5\) in point-slope form is:
\[\boxed{y - 4 = 5(x - 6)}\]

Examples

The point-slope form is a fundamental concept in coordinate geometry used to model linear relationships. For example, if you know the starting temperature of an oven (4 degrees) at a specific time (6 minutes) and the rate at which the oven heats up (5 degrees per minute), you can model the temperature \(y\) at any time \(x\) using the point-slope equation \(y - 4 = 5(x - 6)\). This formula makes it easy to quickly write an equation for a line as soon as you have one point and the slope, which is a common situation in physics and engineering.

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