Math Problem Statement

Write an equation of the line that passes through the given points. (-1,5) and (2,2)

Solution

To find the equation of the line passing through the points (1,5)(-1, 5) and (2,2)(2, 2), we will use the slope-intercept form of a line, y=mx+by = mx + b, where mm is the slope and bb is the y-intercept.

Step 1: Find the slope (mm)

The slope is calculated using the formula: m=y2y1x2x1m = \frac{y_2 - y_1}{x_2 - x_1} Substituting the given points (1,5)(-1, 5) and (2,2)(2, 2): m=252(1)=33=1m = \frac{2 - 5}{2 - (-1)} = \frac{-3}{3} = -1

Step 2: Use the point-slope form

The point-slope form of a line is: yy1=m(xx1)y - y_1 = m(x - x_1) Choose one of the points, for example, (1,5)(-1, 5), and substitute m=1m = -1, x1=1x_1 = -1, and y1=5y_1 = 5: y5=1(x(1))y - 5 = -1(x - (-1)) Simplify: y5=1(x+1)y - 5 = -1(x + 1) y5=x1y - 5 = -x - 1 y=x+4y = -x + 4

Final Equation:

The equation of the line is: y=x+4y = -x + 4

Verification with the second point

Substitute x=2x = 2 into y=x+4y = -x + 4: y=(2)+4=2y = -(2) + 4 = 2 This satisfies the point (2,2)(2, 2).


Let me know if you would like further clarifications! Here are some follow-up questions:

  1. How do you calculate the slope if two points are given?
  2. What is the point-slope form of a line, and how does it relate to slope-intercept form?
  3. Can you derive the line equation if given only one point and the slope?
  4. What happens to the equation if the slope is zero or undefined?
  5. How can we find the intersection point of two lines from their equations?

Tip: Always check the equation by substituting the points to ensure accuracy.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Coordinate Geometry

Formulas

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

Theorems

Equation of a Line

Suitable Grade Level

Grades 8-10