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Chapter 0: Problem 23

Find the slope and \(y\) -intercept. $$2 x+2 y+8=0$$

Short Answer

Expert verified
Slope = -1, y-intercept = -4

Step by step solution

01

- Rewrite the Equation in Slope-Intercept Form

The slope-intercept form of a line is given by: \[ y = mx + b \]First, rewrite the given equation to match this form. Start with the equation:\[ 2x + 2y + 8 = 0 \]Subtract 2x and 8 from both sides to isolate the y-term:\[ 2y = -2x - 8 \]Next, divide both sides of the equation by 2:\[ y = -x - 4 \]Now, the equation is in the slope-intercept form.
02

- Identify the Slope (m)

In the slope-intercept form equation \( y = mx + b \), the coefficient of \( x \) is the slope. Here, \( y = -x - 4 \), so the slope \( m \) is:\[ m = -1 \]
03

- Identify the y-Intercept (b)

In the slope-intercept form equation \( y = mx + b \), the constant term is the y-intercept. Here, \( y = -x - 4 \), so the y-intercept \( b \) is:\[ b = -4 \]

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Key Concepts

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

slope
When we talk about the slope of a line, we're referring to how steep the line is. The slope is a number that describes the direction and steepness of the line. You can think of the slope as the 'rise over run', which means how much the line goes up or down for every horizontal step it takes to the right. For example, in the equation \(y = -x - 4\), the slope is -1. This means for every unit you move to the right (run), you move one unit down (rise). So if you had a point at \( (0,0) \), you'd move right one unit to \( (1,0) \), and from there you would go down to \( (1, -1) \). This negative slope indicates a line decreasing as it moves from left to right.
y-intercept
The y-intercept of a line is the point where the line crosses the y-axis. This is simply the value of y when x is 0. In the slope-intercept form \(y = mx + b\), the y-intercept is represented by the constant term 'b'. Referring to the example equation \(y = -x - 4\), the y-intercept is -4. This means that if you were to graph the line, it would cross the y-axis at the point \( (0, -4) \). The y-intercept helps in graphing the line, as it gives you a starting point.
slope-intercept form
One of the most useful ways to write the equation of a line is in the slope-intercept form, which is \(y = mx + b\). Here, 'm' stands for the slope of the line, and 'b' represents the y-intercept. This form is very handy because it immediately tells you both the slope and where the line crosses the y-axis. For ease of use, you often want to convert equations into this form. If we start with the equation \(2x + 2y + 8 = 0\), we can rewrite it to isolate \(y\) and match the slope-intercept form:
  • Subtract \(2x\) and \(8\) from both sides: \(2y = -2x - 8\)
  • Then divide every term by \(2\) to solve for \(y\): \(y = -x - 4\)
Now the equation is in slope-intercept form, revealing that the slope \(m = -1\) and y-intercept \(-4\).

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