Math Problem Statement

Find the equation of the line that passes through the points (2, 5) and (−3, 6).

Put the equation of the line in the form y  =  mx + b and then enter the values of m and b into the answer box below (in that order), separated with a comma.

Solution

To find the equation of the line that passes through two points (2,5)(2, 5) and (3,6)(-3, 6), we'll use the slope-intercept form of a line:

y=mx+by = mx + b

Step 1: Find the slope mm.

The slope formula between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is:

m=y2y1x2x1m = \frac{y_2 - y_1}{x_2 - x_1}

Substitute the given points (x1,y1)=(2,5)(x_1, y_1) = (2, 5) and (x2,y2)=(3,6)(x_2, y_2) = (-3, 6):

m=6532=15=15m = \frac{6 - 5}{-3 - 2} = \frac{1}{-5} = -\frac{1}{5}

Step 2: Use the slope to find the y-intercept bb.

We now substitute the slope m=15m = -\frac{1}{5} and one of the points into the equation y=mx+by = mx + b to solve for bb. Let's use the point (2,5)(2, 5):

5=15(2)+b5 = -\frac{1}{5}(2) + b

Simplify and solve for bb:

5=25+b5 = -\frac{2}{5} + b b=5+25=255+25=275b = 5 + \frac{2}{5} = \frac{25}{5} + \frac{2}{5} = \frac{27}{5}

Step 3: Write the equation of the line.

Now that we have m=15m = -\frac{1}{5} and b=275b = \frac{27}{5}, the equation of the line is:

y=15x+275y = -\frac{1}{5}x + \frac{27}{5}

Thus, the values of mm and bb are:

m=15,b=275m = -\frac{1}{5}, \quad b = \frac{27}{5}

The final answer is: 1/5,27/5-1/5, 27/5

Would you like further clarification on any step?


Here are 5 follow-up questions:

  1. How do you calculate the slope between any two points?
  2. Why is the slope negative in this case?
  3. What happens if the slope of a line is zero?
  4. How does changing the slope affect the steepness of a line?
  5. Can a line have the same slope but different intercepts?

Tip: Always choose points that are easy to work with when plugging into the slope-intercept formula.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Coordinate Geometry

Formulas

Slope formula: m = (y2 - y1) / (x2 - x1)
Slope-intercept form: y = mx + b

Theorems

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Suitable Grade Level

Grades 8-10