Math Problem Statement

Find the relationship between Input and Output in the given table: Input: [1, 2, 3, 4, 5], Output: [8, 10, 12, 14, 16].

Solution

The table you provided shows the relationship between Input and Output values:

Input12345
Output810121416

Step 1: Observe the pattern

  • The Output increases by 2 each time as the Input increases by 1.
    This indicates a linear relationship.

Step 2: Derive the formula

A linear relationship can be written as:
Output=mInput+b\text{Output} = m \cdot \text{Input} + b where mm is the slope and bb is the y-intercept.

  1. Find the slope mm:
    The slope is calculated as: m=ΔOutputΔInput=10821=2m = \frac{\Delta \text{Output}}{\Delta \text{Input}} = \frac{10 - 8}{2 - 1} = 2

  2. Find the intercept bb:
    Substitute one pair of values, such as (Input=1,Output=8)(\text{Input} = 1, \text{Output} = 8), into the equation:
    8=2(1)+b8 = 2(1) + b
    Solve for bb:
    b=82=6b = 8 - 2 = 6

Final Equation:

The relationship between Input and Output is:
Output=2Input+6\text{Output} = 2 \cdot \text{Input} + 6


Verification:

InputEquation 2Input+62 \cdot \text{Input} + 6Output
12(1)+6=82(1) + 6 = 88
22(2)+6=102(2) + 6 = 1010
32(3)+6=122(3) + 6 = 1212
42(4)+6=142(4) + 6 = 1414
52(5)+6=162(5) + 6 = 1616

Let me know if you'd like me to explain any step in more detail!

Here are 5 related questions to build on this concept:

  1. What is the slope of a line that passes through the points (2,10)(2, 10) and (4,14)(4, 14)?
  2. How do you determine the equation of a line given any two points?
  3. What would the output be if the input is 1010 using this equation?
  4. How can you graph this equation on a coordinate plane?
  5. If the output is 2020, what is the corresponding input?

Tip:

For linear relationships, the slope indicates how much the output changes per unit increase in the input.

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Slope and Intercept
Patterns in Input-Output Relationships

Formulas

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

Theorems

Slope-Intercept Form

Suitable Grade Level

Grade 6-8