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Chapter 2: Problem 35

Exer. 33-36: Find the slope-intercept form of the line that satisfies the given conditions. Through \(A(5,2)\) and \(B(-1,4)\)

Short Answer

Expert verified
The slope-intercept form is \(y = -\frac{1}{3}x + \frac{11}{3}\).

Step by step solution

01

Calculate the slope

First, find the slope of the line passing through points \(A(5,2)\) and \(B(-1,4)\). The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\).Substitute \((x_1, y_1) = (5, 2)\) and \((x_2, y_2) = (-1, 4)\) into the formula:\[m = \frac{4 - 2}{-1 - 5} = \frac{2}{-6} = -\frac{1}{3}.\]
02

Use the point-slope form of the equation

Now, use the slope \(m = -\frac{1}{3}\) and one of the points to write the equation in point-slope form: \(y - y_1 = m(x - x_1)\).Using point \((5, 2)\), substitute in the values:\[y - 2 = -\frac{1}{3}(x - 5).\]
03

Convert to slope-intercept form

Simplify and convert the equation from point-slope form to slope-intercept form, \(y = mx + b\).Start by expanding the equation:\[y - 2 = -\frac{1}{3}x + \frac{5}{3}.\]Add 2 to both sides to solve for \(y\):\[y = -\frac{1}{3}x + \frac{5}{3} + 2.\]Convert 2 to a fraction, \(\frac{6}{3}\), so you can add:\[y = -\frac{1}{3}x + \frac{5}{3} + \frac{6}{3}.\]Combine the fractions:\[y = -\frac{1}{3}x + \frac{11}{3}.\]
04

Verify the equation

Verify your equation \(y = -\frac{1}{3}x + \frac{11}{3}\) by checking it passes through both given points.Substitute \(x = 5\) into the equation and check that \(y = 2\):\[y = -\frac{1}{3}(5) + \frac{11}{3} = -\frac{5}{3} + \frac{11}{3} = \frac{6}{3} = 2.\]Now, substitute \(x = -1\) and check that \(y = 4\):\[y = -\frac{1}{3}(-1) + \frac{11}{3} = \frac{1}{3} + \frac{11}{3} = \frac{12}{3} = 4.\]

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Key Concepts

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

Slope Calculation
Calculating the slope is your first step when determining the equation of a line through two points. The slope describes the steepness and direction of the line. You can calculate the slope by using the formula:
  • \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
Here, \((x_1, y_1)\) and \((x_2, y_2)\) are coordinates of the two points through which the line passes.
For the points \(A(5,2)\) and \(B(-1,4)\), substitute these values into the formula:
  • \( m = \frac{4 - 2}{-1 - 5} = \frac{2}{-6} = -\frac{1}{3} \)
This means that the line decreases by one unit for every three units it moves horizontally to the right. Understanding this calculation provides insight into the behavior of the line.
Point-Slope Form
Once you have the slope, use it with one of the given points to find the line鈥檚 equation in point-slope form. This form is particularly handy because it gives us a straightforward way to incorporate both a point and the slope.
The formula for point-slope form is:
  • \( y - y_1 = m(x - x_1) \)
Substituting point \((5, 2)\) and slope \(m = -\frac{1}{3}\) leads to:
  • \( y - 2 = -\frac{1}{3}(x - 5) \)
This form represents how we can express the line using any point on the line and the slope. It鈥檚 an intermediate step that helps us transition into the slope-intercept form, which is often more intuitive for interpretation.
Equation Verification
Verifying your equation is crucial to ensure it's absolutely correct. For this, you need to check that both given points satisfy the derived equation in slope-intercept form.
The final equation is \( y = -\frac{1}{3}x + \frac{11}{3} \). Plug in the x-coordinates of the original points to see if the resulting y-values match:
  • For \(x = 5\), calculate: \( y = -\frac{1}{3}(5) + \frac{11}{3} = 2 \). This matches \(y = 2\) for point \(A\).
  • For \(x = -1\), calculate: \( y = -\frac{1}{3}(-1) + \frac{11}{3} = 4 \). This matches \(y = 4\) for point \(B\).
Passing through both points confirms the accuracy of your solution. It's like a final check to make sure that no calculation errors slipped through.
Line Through Two Points
A line can uniquely be defined through any two distinct points in a plane. This concept is pivotal in coordinate geometry. Given two points, determining the line's equation involves:
  • Finding the slope, which gives the angle and direction.
  • Using the point-slope form to find a transition equation that bridges to slope-intercept form.
  • Verifying the final equation by confirming that it passes through both points.
This four-step journey from the initial two points to the final equation helps cement your understanding of linear equations. Each step builds upon the last, ensuring a thorough grasp of line equations in a systematic way.

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