Answered

Find the equation of the line passing through each given pair of points:


  1. $(4, 0)$ and $(-3, 3)$
  2. $(6, 7)$ and $(7, 9)$
  3. $(2, -5)$ and $(3, 0)$
  4. $(8, 3)$ and $(-1, -1)$

Verified Answer :

  • Calculate the slope \(m = \frac{y_2 - y_1}{x_2 - x_1}\) for each pair of points.
  • Find the y-intercept \(b = y_1 - mx_1\).
  • Assemble the equation as \(y = mx + b\).
  • The results are:
  1. \((4, 0), (-3, 3) \rightarrow y = -\frac{3}{7}x + \frac{12}{7}\)
  2. \((6, 7), (7, 9) \rightarrow y = 2x - 5\)
  3. \((2, -5), (3, 0) \rightarrow y = 5x - 15\)
  4. \((8, 3), (-1, -1) \rightarrow y = \frac{4}{9}x - \frac{5}{9}\)

The final equations are \(\boxed{y = -\frac{3}{7}x + \frac{12}{7}, y = 2x - 5, y = 5x - 15, y = \frac{4}{9}x - \frac{5}{9}}\).

Explanation

  1. Methodology for Finding Linear Equations

To determine the linear equation \(y = mx + b\) for two given points \((x_1, y_1)\) and \((x_2, y_2)\), we first calculate the slope \(m\) using the formula:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
Once the slope \(m\) is found, we determine the y-intercept \(b\) by substituting one of the points and the slope into the equation \(y = mx + b\), which gives \(b = y_1 - mx_1\). Finally, we write the equation in the form \(y = mx + b\). We will apply this method to the four given pairs of points.

  1. Linear Equation for (4, 0) and (-3, 3)

For the points \((4, 0)\) and \((-3, 3)\):

  1. Calculate the slope: \(m = \frac{3 - 0}{-3 - 4} = \frac{3}{-7} = -\frac{3}{7}\).
  2. Calculate the y-intercept: \(b = 0 - (-\frac{3}{7})(4) = \frac{12}{7}\).
  3. The linear equation is \(y = -\frac{3}{7}x + \frac{12}{7}\).
  1. Linear Equation for (6, 7) and (7, 9)

For the points \((6, 7)\) and \((7, 9)\):

  1. Calculate the slope: \(m = \frac{9 - 7}{7 - 6} = \frac{2}{1} = 2\).
  2. Calculate the y-intercept: \(b = 7 - (2)(6) = 7 - 12 = -5\).
  3. The linear equation is \(y = 2x - 5\).
  1. Linear Equation for (2, -5) and (3, 0)

For the points \((2, -5)\) and \((3, 0)\):

  1. Calculate the slope: \(m = \frac{0 - (-5)}{3 - 2} = \frac{5}{1} = 5\).
  2. Calculate the y-intercept: \(b = 0 - (5)(3) = -15\).
  3. The linear equation is \(y = 5x - 15\).
  1. Linear Equation for (8, 3) and (-1, -1)

For the points \((8, 3)\) and \((-1, -1)\):

  1. Calculate the slope: \(m = \frac{-1 - 3}{-1 - 8} = \frac{-4}{-9} = \frac{4}{9}\).
  2. Calculate the y-intercept: \(b = 3 - (\frac{4}{9})(8) = 3 - \frac{32}{9} = \frac{27 - 32}{9} = -\frac{5}{9}\).
  3. The linear equation is \(y = \frac{4}{9}x - \frac{5}{9}\).

Examples

Linear equations are used in many real-life applications. For example, if you are planning a road trip, the relationship between time traveled and distance covered can be modeled by a linear equation, where the slope represents your average speed and the y-intercept represents your starting distance from the destination. Similarly, businesses use linear models to predict costs based on production volume, where the fixed cost is the y-intercept and the variable cost per unit is the slope. Mastering these equations helps in making accurate predictions and informed decisions.

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