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Chapter 6: Problem 22

Find the equation of the straight line passing through \((-1,4)\) and \((-4,1)\). Does the line pass through \((-2,3)\) ?

Short Answer

Expert verified
The line is \(y = x + 5\), and it passes through \((-2,3)\).

Step by step solution

01

Find the Slope

The slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated using the formula:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]Here, the points are \((-1,4)\) and \((-4,1)\). Substituting these into the formula, we get:\[m = \frac{1 - 4}{-4 - (-1)} = \frac{-3}{-3} = 1\]Hence, the slope \(m\) of the line is 1.
02

Use Point-Slope Form to Write the Equation

With the slope \(m = 1\) and a point on the line, we can use the point-slope form of the equation:\[y - y_1 = m(x - x_1)\]Substituting the point \((-1,4)\) and \(m = 1\):\[y - 4 = 1(x + 1)\]Simplifying gives:\[y = x + 5\]So, the equation of the line is \(y = x + 5\).
03

Verify if the Line Passes Through the Given Point

To check if the line passes through \((-2,3)\), substitute \(x = -2\) into the line's equation \(y = x + 5\):\[y = -2 + 5 = 3\]Since the calculated \(y\) value is equal to the given \(y\) (-2,3), the point \((-2,3)\) lies on the line.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope Calculation
To find the equation of a straight line through two points, we first need to determine the slope. The slope of a line is a measure that describes its steepness and direction. Mathematically, the slope \(m\) is calculated using the formula:
  • \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
Here, \((x_1, y_1)\) and \((x_2, y_2)\) are two distinct points on the line.
By substituting these points into the formula, you can calculate the slope. For the points given in this exercise, \((-1,4)\) and \((-4,1)\), the procedure is as follows:
  • First, calculate the difference in the \(y\) coordinates: \(y_2 - y_1 = 1 - 4 = -3\)
  • Next, find the difference in the \(x\) coordinates: \(x_2 - x_1 = -4 - (-1) = -3\)
  • Finally, divide the differences: \(m = \frac{-3}{-3} = 1\)
The slope \(m = 1\) signifies that for every unit increase in \(x\), \(y\) also increases by one unit. It indicates a perfect diagonal line going upwards from left to right.
Point-Slope Form
Once the slope is known, the point-slope form is a helpful tool for finding the equation of the line. This form is written as:
  • \(y - y_1 = m(x - x_1)\)
This equation represents a line with slope \(m\) passing through the known point \((x_1, y_1)\).
For the exercise, with \(m = 1\) and point \((-1,4)\):
  • Substitute into the formula: \(y - 4 = 1(x + 1)\)
  • Simplify to get \(y = x + 5\)
The equation \(y = x + 5\) describes the line, showing that it crosses the \(y\)-axis at \(5\). It is efficiently derived from one known point and the calculated slope.
Verifying Point on a Line
Verification is a crucial step to ensure the accuracy of our equation. To check if a specific point lies on the line, we substitute its \(x\) coordinate into the line's equation and see if it yields the same \(y\) as given.
  • The given line's equation is \(y = x + 5\).
  • Substitute \(x = -2\) for the point \((-2,3)\): \(y = -2 + 5\)
  • Calculate \(y\), giving \(y = 3\)
Since this derived \(y\) matches the \(y\) coordinate of the original point \((x, y) = (-2, 3)\), it confirms that the point \((-2, 3)\) indeed lies on the line described by the equation \(y = x + 5\). Thus, the work confirms accuracy and completeness of the solution.

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