Math Problem Statement

Match each description to the function that models its linear relationship.

Solution

Let's analyze each description and find the corresponding linear equations from the options provided.

Step-by-Step Solution

  1. First Description: A line goes through points (3,1)(-3, 1) and (7,6)(7, 6).

    • To find the equation, we need to calculate the slope (mm) using the formula: m=y2y1x2x1m = \frac{y_2 - y_1}{x_2 - x_1} Substituting (x1,y1)=(3,1)(x_1, y_1) = (-3, 1) and (x2,y2)=(7,6)(x_2, y_2) = (7, 6): m=617+3=510=12m = \frac{6 - 1}{7 + 3} = \frac{5}{10} = \frac{1}{2}
    • Now, use the point-slope form of the line equation: yy1=m(xx1)y - y_1 = m(x - x_1) Substitute m=12m = \frac{1}{2} and (x1,y1)=(3,1)(x_1, y_1) = (-3, 1): y1=12(x+3)y - 1 = \frac{1}{2}(x + 3) Simplifying further: y=12x+32+1=12x+52y = \frac{1}{2}x + \frac{3}{2} + 1 = \frac{1}{2}x + \frac{5}{2}
    • So, the equation for this line is: y=12x+52y = \frac{1}{2}x + \frac{5}{2}
  2. Second Description: The table with points (0,4)(0, 4) and (5,0)(5, 0).

    • Calculate the slope using the points (0,4)(0, 4) and (5,0)(5, 0): m=0450=45m = \frac{0 - 4}{5 - 0} = -\frac{4}{5}
    • Using the point-slope form with (0,4)(0, 4): y4=45xy - 4 = -\frac{4}{5}x This simplifies to: y=45x+4y = -\frac{4}{5}x + 4
    • The equation for this line is: y=45x+4y = -\frac{4}{5}x + 4
  3. Third Description: A line has a slope of 12-\frac{1}{2} and a y-intercept of 7-7.

    • Since the slope m=12m = -\frac{1}{2} and the y-intercept is 7-7, the equation in slope-intercept form is: y=12x7y = -\frac{1}{2}x - 7

Matching Equations

Now, let's match each description to the equations:

  • First Description (through points (3,1)(-3, 1) and (7,6)(7, 6)): y=12x+52y = \frac{1}{2}x + \frac{5}{2}
  • Second Description (table with points (0,4)(0, 4) and (5,0)(5, 0)): y=45x+4y = -\frac{4}{5}x + 4
  • Third Description (slope 12-\frac{1}{2} and y-intercept 7-7): y=12x7y = -\frac{1}{2}x - 7

Let me know if you need further details or have any questions!


Here are some related questions for further exploration:

  1. How do you determine the y-intercept of a line if only two points are given?
  2. What is the significance of the slope in a linear equation?
  3. How can you quickly identify if a line is increasing or decreasing?
  4. How does changing the slope affect the graph of a line?
  5. What are alternative forms of linear equations, and when are they useful?

Tip: When given two points, always start by finding the slope, then use either point to find the full equation.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations

Formulas

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

Theorems

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Suitable Grade Level

Grades 8-10