Verified Answer :

  • Analyze the provided data points to confirm they lie on a line and identify a linear relationship \(y = mx + b\).
  • Calculate the constant slope \(m\) using two points from the table: \(m = \frac{7 - (-5)}{2 - 0} = 6\).
  • Identify the y-intercept \(b\) as the value of \(y\) when \(x = 0\), which is \(-5\).
  • Combine these to form the linear equation \(\boxed{y = 6x - 5}\).

Explanation

  1. Problem Analysis

The problem asks for a linear equation in the form \(y = mx + b\) that passes through the given points \((-2, -17)\), \((0, -5)\), \((2, 7)\), and \((3, 13)\). A linear relationship has a constant slope \(m\), and the y-intercept \(b\) is the value of \(y\) when \(x = 0\). We have a table of four data points, and we need to determine the constants \(m\) and \(b\) to define the line.

  1. Calculating the Slope

The slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated as:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
Using the points \((0, -5)\) and \((2, 7)\):
\[m = \frac{7 - (-5)}{2 - 0} = \frac{7 + 5}{2} = \frac{12}{2} = 6\]
Thus, the slope of the line is \(m = 6\).

  1. Identifying the Y-Intercept

The y-intercept \(b\) represents the value of \(y\) when \(x = 0\). From the given table, when \(x = 0\), the corresponding value for \(y\) is \(-5\). Therefore, the y-intercept is \(b = -5\). Alternatively, using the slope \(m = 6\) and the point \((2, 7)\) in the equation \(y = mx + b\), we get:
\[7 = 6(2) + b\]
\[7 = 12 + b\]
\[b = 7 - 12 = -5\]
Both methods confirm that the y-intercept is \(-5\).

  1. Forming and Verifying the Equation

With the slope \(m = 6\) and the y-intercept \(b = -5\), the equation of the line is \(y = 6x - 5\). To ensure this equation is correct for all points, let us verify it with the remaining data:
For \((-2, -17)\): \(y = 6(-2) - 5 = -12 - 5 = -17\). This matches the table.
For \((3, 13)\): \(y = 6(3) - 5 = 18 - 5 = 13\). This also matches the table.
The equation that fits all provided points is \(\boxed{y = 6x - 5}\).

Examples

Linear equations like \(y = 6x - 5\) are fundamental in modeling real-world data that changes at a constant rate. For example, if you are filling a pool, \(x\) could represent the time in hours and \(y\) could be the depth of the water in inches, starting from an initial level. Understanding how to find this equation allows you to predict future values—such as how deep the water will be after 5 hours—by simply calculating \(y = 6(5) - 5 = 25\) inches. This makes linear modeling an essential tool for forecasting and planning in various fields, from science to personal finance.

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