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Chapter 1: Problem 64

Find an equation of the line that satisfies the given conditions. Through \(A(5,2)\) and \(B(-1,4)\)

Short Answer

Expert verified
The line's equation is \(y = -\frac{1}{3}x + \frac{11}{3}\).

Step by step solution

01

Find the Slope of the Line

To find the slope (m) of the line through points \(A(5, 2)\) and \(B(-1, 4)\), use the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the coordinates of \(A\) and \(B\): \[ m = \frac{4 - 2}{-1 - 5} = \frac{2}{-6} = -\frac{1}{3} \] Thus, the slope of the line is \(-\frac{1}{3}\).
02

Use Point-Slope Form to Write the Equation

With the slope \(m = -\frac{1}{3}\) and point \(A(5, 2)\), apply the point-slope form of the equation: \[ y - y_1 = m(x - x_1) \] Substitute \((x_1, y_1) = (5, 2)\) and \(m = -\frac{1}{3}\): \[ y - 2 = -\frac{1}{3}(x - 5) \] This is the equation in point-slope form.
03

Convert to Slope-Intercept Form

Simplify the point-slope equation to get it into slope-intercept form \(y = mx + b\). Start with the equation from step 2: \[ y - 2 = -\frac{1}{3}(x - 5) \] Distribute the slope: \[ y - 2 = -\frac{1}{3}x + \frac{5}{3} \] Add 2 to both sides: \[ y = -\frac{1}{3}x + \frac{5}{3} + 2 \] Convert 2 to \(\frac{6}{3}\) and add: \[ y = -\frac{1}{3}x + \frac{11}{3} \] So, the equation in slope-intercept form is \(y = -\frac{1}{3}x + \frac{11}{3}\).

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Key Concepts

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

Understanding Slope
The slope of a line is a way to describe its steepness or tilt. Mathematically, the slope (\( m \)) is determined by the ratio of the change in the\( y \)-coordinate to the change in the\( x \)-coordinate between two points on the line. This can be represented by the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
  • The numerator\( y_2 - y_1 \)is the change in the vertical direction.
  • The denominator\( x_2 - x_1 \)is the change in the horizontal direction.
For example, using points\( A(5, 2) \)and\( B(-1, 4) \)we calculate:
\[ m = \frac{4 - 2}{-1 - 5} = \frac{2}{-6} = -\frac{1}{3} \]
Hence, the slope is \(-\frac{1}{3}\)indicating a line that falls as it moves to the right.
Point-Slope Form
The point-slope form of a linear equation is particularly useful when you know the slope of a line and one point on the line. It is given by:
\[ y - y_1 = m(x - x_1) \]
  • \( m \)is the slope calculated from two known points on the line.
  • \( (x_1, y_1) \)is the point through which the line passes.
Using point A\( (5, 2) \)and the slope \(-\frac{1}{3}\)we get:
\[ y - 2 = -\frac{1}{3}(x - 5) \]
This equation represents the line in point-slope form and can be directly used to plot the line or transition into other line forms.
From Point-Slope to Slope-Intercept Form
Converting the point-slope form to the slope-intercept form allows for a clear illustration of how the line crosses the y-axis. Slope-intercept form is given by:
\[ y = mx + b \]
  • \( m \)is the slope.
  • \( b \)is the y-intercept, where the line crosses the y-axis.
Starting with the point-slope equation\( y - 2 = -\frac{1}{3}(x - 5) \)we need to rearrange it:
1. Distribute the slope:\( y - 2 = -\frac{1}{3}x + \frac{5}{3} \)
2. Add 2 to both sides to solve for\( y \):\[ y = -\frac{1}{3}x + \frac{5}{3} + 2 \]
3. Convert 2 into a fraction:\( \frac{6}{3} \)and add it:\[ y = -\frac{1}{3}x + \frac{11}{3} \]
Thus, the slope-intercept form is\( y = -\frac{1}{3}x + \frac{11}{3} \)which clearly shows the slope and where it crosses the y-axis.

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