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Chapter 3: Problem 12

Find the slope and the \(y\) -intercept of the line with the given equation. $$y=5-x$$

Short Answer

Expert verified
The slope of the line is '-1' and the y-intercept is '5'.

Step by step solution

01

Identify the slope

The equation given is \(y = 5 - x\). In this equation, the coefficient of \(x\) is '-1'. Therefore, the slope of the line (\(m\)) is '-1'.
02

Identify the y-intercept

The y-intercept is the value of \(y\) when \(x = 0\). In our equation, it is the constant term. Hence, the y-intercept (\(c\)) is '5'.

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Key Concepts

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

Understanding the Slope
The slope of a line is a measure of its steepness. It's represented by the letter "m" in the equation of a line in the form \( y = mx + c \). The slope describes how much the y-coordinate of a point on the line changes for a one-unit increase in the x-coordinate.
For example:
  • If the slope is positive, the line rises as you move from left to right.
  • If the slope is negative, the line falls as you go from left to right.
  • If the slope is zero, the line is horizontal.
  • If the slope is undefined, the line is vertical.
In the example equation \( y = 5 - x \), we can rewrite it in standard slope-intercept form as \( y = -1x + 5 \). Hence, the slope "m" is \(-1\), meaning the line decreases by one unit vertically for every one unit it moves horizontally.
Grasping the Y-intercept
The y-intercept of a line is the point where the line crosses the y-axis. It indicates the value of \( y \) when \( x \) is zero. In the standard linear equation form \( y = mx + c \), the y-intercept is represented by the constant "c."
Understanding the y-intercept helps to determine where the line will intersect the vertical axis on a graph.
  • A positive y-intercept shifts the line upwards.
  • A negative y-intercept shifts the line downwards.
For the equation \( y = 5 - x \), setting \( x = 0 \) results in \( y = 5 \). This tells us that the line crosses the y-axis at \( y = 5 \). Thus, the y-intercept in this case is 5.
Equation of a Line
The equation of a line in two-dimensional space fundamentally relates \( x \) and \( y \). It can predict or describe points on the line. The most common forms of line equations include the slope-intercept form \( y = mx + c \), and the point-slope form \( y - y_1 = m(x - x_1) \).
In the slope-intercept form:
  • "m" represents the slope.
  • "c" represents the y-intercept.
In our example \( y = 5 - x \), which is the slope-intercept form \( y = -1x + 5 \):
  • "-1" is the slope, reflecting the line's downward angle.
  • "5" is the y-intercept, indicating where the line crosses the y-axis.
Equations are vital tools in graphing and analyzing lines in coordinate systems.

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Most popular questions from this chapter

Solve: \(\quad 8(x-2)-2(x-3) \leq 8 x\).

What is a \(y\)-intercept of a graph?

Use a graphing utility to graph in a standard viewing rectangle, \([-10,10,1]\) by \([-10,10,1]\). Then use the \([\text { TRACE }]\) feature to trace along the line and find the coordinates of two points. $$y=-3 x+2$$

$$\text { Simplify: } \quad 3\left(12 \div 2^{2}-3\right)^{2}$$.

Solve for \(h: \quad V=\frac{1}{3} A h .\) (Section 2.4, Example 4)
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