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Chapter 1: Problem 36

Find an equation of the line that satisfies the given conditions. (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
The line equation is \(y = -2x + 7\).

Step by step solution

01

Understand the slope-intercept form

The slope-intercept form of a line's equation is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
02

Identify the slope

From the problem, the slope \( m \) is given as \(-2\).
03

Plug in the point into the equation

We know the line passes through the point \((4, -1)\). Substitute \(x = 4\), \(y = -1\), and \(m = -2\) into the slope-intercept equation: \(-1 = -2 \times 4 + b\).
04

Solve for the y-intercept \(b\)

Calculate \(-2 \times 4 = -8\), then substitute: \(-1 = -8 + b\). Solve for \(b\) by adding 8 to both sides: \(b = 7\).
05

Write the final equation of the line

Now that we know \(m = -2\) and \(b = 7\), substitute these back into the slope-intercept form: \(y = -2x + 7\). This is the equation of the line.

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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 linear equation is a popular way to express lines in algebra, especially when you want to identify the slope and the y-intercept quickly. Imagine it as a template: \( y = mx + b \). Here, \( m \) signifies the slope, and \( b \) represents the y-intercept. This form is very useful because it allows you to spot how steep the line is and where it touches the y-axis almost at first glance.

This format makes graphing and understanding the behavior of lines straightforward:
  • The slope \( m \) tells you how steep the line rises or falls as you move along the x-axis.
  • The y-intercept \( b \) is the point where the line crosses the y-axis, providing a clear reference point.
By using this form, creating and analyzing equations of lines becomes less of a mystery. It's like having a map that points out the direction and where it begins!
Slope
The slope of a line is its steepness, and it's represented by the variable \( m \) in the slope-intercept form. Think of it as the tilt or incline of the line. If you visualize a hill, the slope indicates how steep the hill is. In mathematical terms, it describes the change in the y-coordinate as the x-coordinate changes.

Here's how the slope works:
  • If the slope is positive, the line goes upwards as you move from left to right.
  • If the slope is negative, like in our exercise (-2), the line slants downward.
  • A slope of zero means the line is completely flat, having no vertical change as you move horizontally.
To find the slope between two points, imagine the formula \( \frac{\Delta y}{\Delta x} \), which shows the ratio of vertical change to horizontal change. Essentially, it tells you how many units the line rises or falls for every unit you move to the right.
Y-Intercept
The y-intercept is where a line crosses the y-axis, represented by the "\( b \)" in the slope-intercept form \( y = mx + b \). It's like the starting point of the line when you are graphing it on a coordinate plane.

When a line crosses the y-axis, this means that \( x = 0 \). By substituting \( x = 0 \) into the equation, the y-value you find is precisely the y-intercept:
  • In the example from the exercise, solving \(-1 = -8 + b\) helped us find \( b = 7 \).
  • This means that the line crosses the y-axis at \( y = 7 \).
Knowing the y-intercept lets you start plotting a graph easily. From this point, you can follow the direction given by the slope to sketch the whole line. It essentially grounds the line to the graph by providing a definitive point where it must pass.

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Most popular questions from this chapter

Factor the expression by grouping terms. $$2 x^{3}+x^{2}-6 x-3$$

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