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Chapter 2: Problem 36

(a) Sketch the line with slope \(-2\) that passes through the point \((4,-1)\) . (b) Find an equation for this line.

Short Answer

Expert verified
Equation: \( y = -2x + 7 \)

Step by step solution

01

Understand the Slope-Intercept Form

The slope-intercept form of the equation of a line is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Our objective is to find this form of the equation for the line with a slope of \(-2\) and passing through the point \((4, -1)\).
02

Plug in the Known Values

We have a slope \( m = -2 \) and a point \((x_1, y_1) = (4, -1)\) through which the line passes. Substitute these into the formula of a line to get \( -1 = -2(4) + b \).
03

Solve for b, the Y-Intercept

Calculate \(-2(4)\), which gives \(-8\). Then substitute back: \( -1 = -8 + b \). Solve for \( b \) by adding \( 8 \) to both sides to find \( b = 7 \).
04

Write the Equation of the Line

Now that we know \( b = 7 \), we can write the equation of the line as \( y = -2x + 7 \).
05

Sketch the Line

To sketch the line, plot the y-intercept \((0, 7)\) and use the slope of \(-2\) to find another point. Starting at \((0, 7)\), move down 2 units and right 1 unit to reach \((1, 5)\). Draw a line through these points extending in both directions, and check that it passes through \((4, -1)\).

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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 is a cornerstone concept in understanding linear equations. It's given by the formula \( y = mx + b \). Let's break this down:
  • \(y\) represents the dependent variable, often referring to the vertical axis.
  • \(x\) is the independent variable, usually associated with the horizontal axis.
  • \(m\) is the slope, which describes the steepness and direction of the line.
  • \(b\) represents the y-intercept, which is where the line crosses the y-axis.
In our exercise, the slope \(m\) is given as \(-2\). We need to fit this into the equation format by discovering the appropriate value for \(b\) so we can fully describe our line.
Finding Y-Intercept
Finding the y-intercept means identifying the point where the line crosses the y-axis, which is crucial in writing the full equation for a line. When a point is given, such as \(4,-1\), it helps us pinpoint that value:
  • Starting with the slope-intercept form: \( y = mx + b \).
  • Substitute the point (4, -1) into the equation: \( -1 = -2 \cdot 4 + b \).
  • This simplifies to: \( -1 = -8 + b \).
  • To solve for \(b\), add \(8\) to both sides: \( b = 7 \).
Thus, the y-intercept of our line is \(7\), completing our equation as \( y = -2x + 7 \). This step ensures our line accurately represents both its slope and intersection with the y-axis.
Graphing Lines
Graphing lines with precision involves using both the slope and y-intercept. Here's how it works:
  • Begin by plotting the y-intercept on the graph. In our case, it's at \(0, 7\).
  • The slope \(-2\) indicates the line goes down two units for every unit it goes right. From the y-intercept, move down 2 units and right 1 unit to locate a new point \(1, 5\).
  • Each new point helps define the line. Continue plotting using the slope if needed to ensure accuracy.
  • Once you have two or more points, draw a line through them, extending in both directions.
Verifying your line with the original point, such as confirming it passes through \(4, -1\), is a great way to ensure its correctness. Understanding graphing lines in this manner solidifies your grip on linear relationships in a visual format.

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Most popular questions from this chapter

Find a point on the \(y\) -axis that is equidistant from the points \((5,-5)\) and \((1,1) .\)

Suppose you are given the coordinates of three points in the plane, and you want to see whether they lie on the same line. How can you do this using slopes? Using the Distance Formula? Can you think of another method?

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Use slopes to show that \(A(1,1), B(7,4), C(5,10),\) and \(D(-1,7)\) are vertices of a parallelogram.
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