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Chapter 1: Problem 20

Find the equation of the line through the given pair of points. Solve it for \(y\) if possible. $$(-2,1),(3,5)$$

Short Answer

Expert verified
The equation of the line is \(y = \frac{4}{5}x + \frac{13}{5}\).

Step by step solution

01

Identify the points

The given points are \((-2, 1)\) and \(3, 5)\). Name them as point \(A(-2, 1)\) and point \(B(3, 5)\).
02

Find the slope

Use the formula for the slope, \(m\), which is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the values from points \(A\) and \(B\), we get: \[ m = \frac{5 - 1}{3 - (-2)} = \frac{4}{5} \]
03

Write the slope-intercept form

The slope-intercept form of a line is: \[ y = mx + b \] We know the slope \(m = \frac{4}{5}\). Now, plug one of the points (let's use point \(-2, 1\)) to find \(b\): \[ 1 = \frac{4}{5}(-2) + b \]
04

Solve for the y-intercept (b)

Solve the equation: \[ 1 = -\frac{8}{5} + b \] \[ b = 1 + \frac{8}{5} = \frac{5}{5} + \frac{8}{5} = \frac{13}{5} \]
05

Write the final equation

Substitute the values of \(m\) and \(b\) into the slope-intercept form: \[ y = \frac{4}{5}x + \frac{13}{5} \]

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form of a line is beneficial for quickly identifying the slope and y-intercept of a straight line. \( y = mx + b \)
  • Here, \m\is the slope of the line, and \b\ is the y-intercept.
Step 3 in the problem involves substituting the slope and one of the points into the slope-intercept formula to find \b\.
For example, using point \ (-2, 1) \:

  • Substitute 1 for \ y \, 4/5 for \ m \, and -2 for \ x \:
    \((1 = \frac{4}{5} (-2) + b) \)

Then solve for \b\ (the y-intercept), and you are halfway to writing the equation of the line.
Y-Intercept
The y-intercept is the point where the line crosses the y-axis. To find this, plug the slope and one point's coordinates into the slope-intercept form of the line.
From the slope-intercept form: \( y = mx + b \), already identified in the previous steps.
  • Substitute the point \( ( -2, 1 ) \) and the slope \( \ 4/5 \)

    \ (( 1= \frac{4}{5}(-2) + b) \)
  • Solve for \ b \ which is the y-intercept: \ (b = 1 + \frac{8}{5}) \
  • \ b = \frac{13}{5} \
Now substitute the slope and y-intercept back into the slope-intercept equation to get the line's full equation: \(( y = \frac{4}{5}x + \frac{13}{5}) \

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Most popular questions from this chapter

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