Math Problem Statement

Write the slope-intercept form of the equation of a line passing through the points (2,5) and (1,8).

Solution

To find the equation of a line passing through the points (2,5)(2,5) and (1,8)(1,8) in slope-intercept form, y=mx+by = mx + b, we first need to determine the slope mm and then the y-intercept bb.

Step 1: Calculate the Slope mm

The formula for the slope 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}

For the points (2,5)(2, 5) and (1,8)(1, 8): m=8512=31=3m = \frac{8 - 5}{1 - 2} = \frac{3}{-1} = -3

Step 2: Substitute a Point to Find bb

Now that we know m=3m = -3, we can use one of the points to find bb. Let's use (2,5)(2,5):

y=mx+b5=3(2)+by = mx + b \Rightarrow 5 = -3(2) + b 5=6+b5 = -6 + b b=11b = 11

Step 3: Write the Equation

With m=3m = -3 and b=11b = 11, the equation of the line is: y=3x+11y = -3x + 11

Conclusion

The correct answer is the first option: y=3x+11y = -3x + 11.

Would you like further details on the steps, or do you have any questions?


Here are some related questions for further practice:

  1. How do you determine the slope of a line given two points?
  2. How do you convert a point-slope form equation to slope-intercept form?
  3. What is the difference between slope-intercept form and point-slope form?
  4. How do you find the equation of a line parallel to a given line?
  5. How do you find the equation of a line perpendicular to a given line?

Tip: When given two points, always start by calculating the slope. It helps set the foundation for finding the equation of the line.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Slope-Intercept Form

Formulas

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

Theorems

Slope and y-intercept of a line

Suitable Grade Level

Grade 8