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Chapter 2: Problem 16

Write an equation in slope-intercept form of the line that satisfies the given conditions. See Example 1. $$m=2 ; b=12$$

Short Answer

Expert verified
The equation is \(y = 2x + 12\).

Step by step solution

01

Understand the Slope-Intercept Form

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

Identify the Given Values

From the exercise, it is given that the slope \(m = 2\) and the y-intercept \(b = 12\).
03

Substitute the Values into the Equation

Substitute \(m = 2\) and \(b = 12\) into the slope-intercept form equation \(y = mx + b\): \[y = 2x + 12\]

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Key Concepts

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

Linear Equation
A linear equation represents a straight line on a graph. The general format for a linear equation is y = mx + b.
This form is known as the slope-intercept form. Each component of this equation has a specific role to play.
The key features of a linear equation are:
  • y: The dependent variable or the output value that we are trying to find or predict.
  • x: The independent variable or the input value that we control or manage.
  • m: The slope of the line, showing how steep the line is. It indicates the rate of change of y concerning x.
  • b: The y-intercept, which is where the line intersects the y-axis.

In our exercise, we were given m = 2 and b = 12. By substituting these values into the slope-intercept form, we created the equation y = 2x + 12.Understanding these components helps to visualize how changes in x affect y and vice versa.
Slope
In the slope-intercept form of a linear equation, the slope is represented by the letter m. The slope tells us how steep a line is.
It is calculated as the rise over run, which means the change in the y-value for a one-unit change in the x-value.
  • For example, in our equation y = 2x + 12, the slope m = 2 means that for every increase of 1 in x, y increases by 2.


m can be positive, negative, zero, or undefined.
  • If m is positive, the line slopes upward from left to right.
  • If m is negative, the line slopes downward from left to right.
  • If m is zero, the line is horizontal, meaning y does not change as x changes.
  • If m is undefined, the line is vertical, meaning x does not change as y changes.

Understanding the slope is crucial because it helps to predict and understand the relationship between the two variables.
Y-Intercept
The y-intercept is the point where the line crosses the y-axis. In the slope-intercept form equation, it is represented by the letter b.
At the y-intercept, the value of x is zero.
In our given problem, the y-intercept b was 12, which means the line crosses the y-axis at the point (0, 12).
This can be visualized by plugging x = 0 into the equation y = 2x + 12 and solving for y.
y = 2(0) + 12 = 12
  • The y-intercept is essential for graphing because it provides a starting point for drawing the line. Once the y-intercept is known, the slope can be used to find other points on the line by moving from the y-intercept.

Knowing the y-intercept helps in understanding the initial value in real-world situations.

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