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Algebra Foundations

You will write the equation of a line given the slope and y-intercept, the slope and a point, or only two points.

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After completing this tutorial, you will be able to complete the following:

- Write the equation of a line in slope-intercept form given the slope and y-intercept.
- Write the equation of a line in slope-intercept form given the slope and a point.
- Write the equation of a line in slope-intercept form given two points.

Linear relationships exist in a wide variety of mathematical, scientific, and ordinary real-world circumstances. This fact is due to the prevalence of constant rates and proportionality in many mathematically describable situations. Varying quantities are linearly related just in case there is a constant ratio of change between the quantities. In many such relationships, such as the relationship between the quantities of income and wage, one quantity can be viewed as dependent on the other. The variables x and y are frequently (but certainly not always) used in these scenarios; x represents the independent variable, and y represents the dependent variable. Specific values of the dependent variable are uniquely determined by values of the independent variable. With this vocabulary, we say that there is a linear relationship between x and y if there is a constant ratio between the change in y and the change in x. The constant ratio of change is called the rate of change.

Linear relationships can be expressed by a variety of methods; here we focus on graphs and equations. Graphically, two-variable linear relationships can be described by straight lines in the plane.

The above graph depicts a relationship between x and y. Each point on the line corresponds to a pair of values in the relationship between x and y. For example, the points (0,1) and (1,3) are on the line, so that y=1 when x=0 and y=3 when x=1. These values can be used to find the rate by which y changes with respect to x.

The slope of a line, a measure of the line's steepness, coincides with the rate of change. Recall the slope formula, which is used to find the slope of a line given two points on the line.

Each line has two distinguished points at which the line crosses the x- and y-axes. These are called the x- and y-intercept, respectively. The y-intercept can be viewed as an initial value, since it corresponds to x = 0 in the underlying relationship.

A line's slope and intercept can be used to write an algebraic equation that describes the line. This equation also describes the underlying linear relationship, so such equations are called linear equations. One line can be described with infinitely many equivalent linear equations, so a handful of commonly occurring forms are distinguished. One of these is the slope-intercept form,y = mx + b, where m is the line's slope and b is the y-intercept. This is the most commonly used form of a linear equation, owing in part to its usefulness. The slope and y-intercept of a line, interpreted as rate of change and initial value for the underlying linear relationship, provide important information about the relationship. Geometrically, these values allow us to easily visualize a line and draw its graph. Moreover, the slope and y-intercept are complete invariants in the sense that they uniquely determine each and every line.

Writing the equation of a line in slope-intercept form can be done by using a variety of methods, the suitability of each depends on the given information. All of these methods follow the same general framework:

1) Identify the slope.

2) Identify the y-intercept.

3) Write the equation in slope-intercept form.

The most common situations are to find the slope of a line given any of the following information:

· Slope and y-intercept

· Slope and a point

· Two points on the line

For example, to find the equation of the line that passes through the points (1, 2) and (2, 3), the slope formula is used to find the line's slope, which is represented by the letter m.

Approximate Time | 20 Minutes |

Pre-requisite Concepts | Students should know how to find the slope of an equation, graph linear equations, identify a linear equation, interpret the meaning of a slope in the context of a linear model, recognize a linear equation in slope-intercept form, and solve a linear equation. |

Course | Algebra Foundations |

Type of Tutorial | Skills Application |

Key Vocabulary | equation, equation of a line, linear equation |