Math Problem Statement

Below is a graph that contains the lines for all the above equations. Find the line that represents each equation, based on the ordered pairs, and write the equation next to each letter for the line that it represents.

Solution

To solve this problem, we need to determine the equations of the lines labeled A, B, C, D, and E based on the graph provided.

Step-by-Step Process:

  1. Identify two points on each line: From the graph, pick two points for each labeled line that are easy to read, like points where the line crosses grid intersections.

  2. Use the slope formula:
    m=y2y1x2x1m = \frac{y_2 - y_1}{x_2 - x_1} The slope mm describes how steep the line is.

  3. Determine the y-intercept: This is the point where the line crosses the yy-axis (where x=0x = 0).

  4. Form the equation: The equation of a line can be written in the form: y=mx+by = mx + b Where mm is the slope and bb is the y-intercept.

Let’s begin by analyzing the points on the graph for each line:

Line A:

  • Points: (7,0)(-7, 0) and (0,2)(0, 2)
  • Slope mAm_A: mA=200(7)=27m_A = \frac{2 - 0}{0 - (-7)} = \frac{2}{7}
  • Y-intercept bA=2b_A = 2
  • Equation of Line A: y=27x+2y = \frac{2}{7}x + 2

Line B:

  • Points: (1,0)(1, 0) and (0,1)(0, 1)
  • Slope mBm_B: mB=1001=1m_B = \frac{1 - 0}{0 - 1} = -1
  • Y-intercept bB=1b_B = 1
  • Equation of Line B: y=x+1y = -x + 1

Line C:

  • Points: (2,0)(2, 0) and (0,1)(0, 1)
  • Slope mCm_C: mC=1002=12m_C = \frac{1 - 0}{0 - 2} = -\frac{1}{2}
  • Y-intercept bC=1b_C = 1
  • Equation of Line C: y=12x+1y = -\frac{1}{2}x + 1

Line D:

  • Points: (4,0)(4, 0) and (0,2)(0, 2)
  • Slope mDm_D: mD=2004=12m_D = \frac{2 - 0}{0 - 4} = -\frac{1}{2}
  • Y-intercept bD=2b_D = 2
  • Equation of Line D: y=12x+2y = -\frac{1}{2}x + 2

Line E:

  • Points: (0,5)(0, -5) and (5,0)(5, 0)
  • Slope mEm_E: mE=0(5)50=55=1m_E = \frac{0 - (-5)}{5 - 0} = \frac{5}{5} = 1
  • Y-intercept bE=5b_E = -5
  • Equation of Line E: y=x5y = x - 5

Final Equations:

  • Line A: y=27x+2y = \frac{2}{7}x + 2
  • Line B: y=x+1y = -x + 1
  • Line C: y=12x+1y = -\frac{1}{2}x + 1
  • Line D: y=12x+2y = -\frac{1}{2}x + 2
  • Line E: y=x5y = x - 5

Do you want further details on any specific part of this process?


Here are some related questions:

  1. How can we verify the correctness of the slope calculations?
  2. What happens to the equation if the slope is 0 or undefined?
  3. How can we use these equations to find intersections between the lines?
  4. How does the y-intercept affect the position of the line?
  5. Can you derive the equation of a vertical line from this graph?

Tip: Remember, the y-intercept is always the value of yy when x=0x = 0.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Graphing

Formulas

Slope formula: m = (y2 - y1) / (x2 - x1)
Equation of a line: y = mx + b

Theorems

Slope-Intercept Form
Line Equation Theorem

Suitable Grade Level

Grades 8-10