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Chapter 2: Problem 15

An equation written in the form \(y=m x+b\) is said to be written in _____- ______ form.

Short Answer

Expert verified
The equation is written in 'slope-intercept form'.

Step by step solution

01

Identify the Equation Form

Analyze the given equation format, which is written as: \( y = mx + b \)
02

Define Each Component

Recognize that in the equation \( y = mx + b \), 'm' represents the slope of the line and 'b' represents the y-intercept, which is the point where the line crosses the y-axis.
03

Name the Equation Form

The equation \( y = mx + b \) is known as the 'slope-intercept form' because it explicitly shows the slope and the y-intercept of the line.

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Key Concepts

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

linear equations
A linear equation is an equation that makes a straight line when it is graphed on a coordinate plane. It often looks like this: \(y = mx + b\).

This simple equation has two main parts:
  • The variable 'y' represents the dependent variable. It's what you find out after plugging 'x' into the equation.
  • The variable 'x' is the independent variable. It鈥檚 what you choose values for to find out corresponding values of 'y'.
Linear equations can have other forms besides the slope-intercept form, such as the standard form \(Ax + By = C\). No matter the form, if there are no exponents or special functions (like sine or cosine), the equation will graph as a straight line.
slope
The slope is a measure of how steep a line is. In the equation \(y = mx + b\), 'm' stands for the slope. The slope tells us how much 'y' changes for a change in 'x'.

If the slope is positive, the line slants upwards from left to right. If it鈥檚 negative, the line slants downwards. A larger slope means a steeper line.

To calculate the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\), use the formula:
\[ m = \frac{(y_2 - y_1)}{(x_2 - x_1)} \]This represents the 'rise' over the 'run', or how much 'y' 'rises' over the change in 'x'.
y-intercept
The y-intercept is the value where the line crosses the y-axis. This happens when 'x' is zero. In the equation \(y = mx + b\), 'b' stands for the y-intercept.

When you plot a line, the y-intercept is the starting point before considering the slope.
  • So if your equation is \(y = 2x + 3\), it crosses the y-axis at (0,3).
  • The y-intercept helps anchor the line in a specific spot before it stretches up or down with the slope.
Understanding the y-intercept gives you a fixed point to start graphing any linear equation.
graphing lines
Graphing lines involves plotting points and drawing a straight line that matches the equation. With the slope-intercept form \(y = mx + b\), it's especially easy.

Here鈥檚 how to do it:
  • Start by plotting the y-intercept 'b' on the y-axis.
  • Use the slope 'm' to determine the next points: from the y-intercept, rise and run according to the ratio given by the slope.
For example:
  • If the equation is \(y = 2x + 1\), plot the point (0,1).
  • From (0,1), move up 2 units and right 1 unit to find the next point at (1,3).
Connect these points to make your line. Repeat this process for accuracy and ensure your line perfectly represents the equation.

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