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The slope-intercept form of the linear equation

The slope-intercept form of an equation of the first degree is a way of expressing that equation in the form of the equation of a straight line . In other words, it is expressed with the same mathematical form as a function that, when graphed in a Cartesian coordinate system, results in a straight line. A linear equation expressed in this way has the following mathematical form:

equation of line in slope-intercept form

As can be seen, this way of representing linear equations is characterized by having the variable that we commonly consider as the dependent variable (in most cases and , although this can vary) isolated in one of the members of the equation (usually the left) with coefficient 1; while the other member is composed of a term that contains the independent variable (usually x ) and an independent term.

Interpretation of the linear equation in slope-intercept form

When expressed in this way, the coefficient of the independent variable, in this case m , represents the slope of the line when this equation is graphed in a Cartesian coordinate system.

On the other hand, the independent term, in this case b , indicates the point at which the line cuts or intersects the ordinate axis or y axis, as shown in the following graph. That is precisely why it is called slope-intercept form.

shape slope intersection

Slope interpretation

The slope ( m ) indicates how much the value of y of a point on the line changes by increasing the value of x by one unit , thus it represents the slope of the line. This value can be any rational number, both positive and negative. There are three possible ranges of values ​​that are interpreted differently:

  • A positive slope (m>0) indicates that the line goes up as we move from left to right on the graph.
  • When the independent variable term does not appear (that is, when there is no x in the equation) it means that the slope is zero (m=0). In this case, the line is horizontal or parallel to the abscissa axis (x-axis).
  • When the slope is negative (m<o), the line goes down as we move from left to right on the graph.

Interpretation of the intersection

The independent term, b , represents the intersection point of the line with the ordinate axis, that is, with the y axis in the Cartesian coordinate system. In those cases in which there is no independent term, it is understood that its value is zero (b=0) so the line passes through the origin of the coordinate system.

Special cases of the equation of a line in slope-intercept form

Case 1: y = b

slope-intercept shape with slope 0

When the equation has the previous form, that is, when the term of the independent variable does not appear, it is understood that the slope is zero and that, therefore, the equation represents a horizontal line that passes through the point (0;b ).

Case 2: y = mx

positive slope slope-intercept shape

When there is no independent term, it means that its value is zero, and therefore, it intersects the y-axis at 0. This means that the line passes through the origin of the coordinate system.

Case 3: 0 = mx + b

slope-intercept shape with undefined slope

In this case, it consists of a vertical line (parallel to the y axis) that intersects the abscissa axis (or x axis) at the point x = – b/m, as shown in the previous graph.

This is an unusual form of the equation of a line in which the coefficient m and the independent term b lose their normal meaning. A vertical line has an undefined slope, that is, its slope does not exist. This is not the same as saying that its slope is zero.

On the other hand, since it is a vertical line parallel to the y axis, it never intersects that axis. Therefore, the independent term, b, no longer indicates the intersection as it did in the previous cases.

Advantages of the slope-intercept form

Compared to the other ways of representing linear equations, the slope-intercept form has the following advantages:

  • Immediately returns the values ​​of the slope and the y-intercept of the line.
  • The above allows to visualize in a very simple and fast way the graph of a linear equation in a Cartesian coordinate system.
  • By providing the value of the slope, it allows you to quickly calculate the angle that the line makes with the x-axis using the tangent.
  • It allows you to quickly know if two lines are parallel to each other or not, simply by comparing their slopes.
  • It allows you to quickly determine whether or not two lines are perpendicular to each other.
  • Just looking at the form of the equation lets us know immediately if it is an increasing, decreasing, horizontal, or vertical line.
  • Lets you calculate the y-coordinate of any point on the line given its x-value in one step.
  • It facilitates the substitution method for solving systems of linear equations of two variables because the equation is already solved for one of them (y).

Steps to transform standard form to slope-intercept form

In addition to slope-intercept form, the equation of a line can also be represented in other ways, the most important of which is standard form:

general shape

In this case, the coefficients A, B, and C are integers. When you have an equation expressed in this way and you want to write it in slope-intercept form, you only have to follow the following steps:

Step 1: Ax is subtracted from both sides of the equation.

Step 2: all the coefficients and the independent term are divided by the coefficient B (including its sign).

Step 3: If possible, simplify any fraction that arose from the division.

Examples of transformation from standard form to slope-intercept form

Example 1: 3x + 2y = 4

Step 1:

Example of slope-intercept form

Step 2:

Example of slope-intercept form

Step 3:

Example of slope-intercept form

As you can see, this equation corresponds to a descending line that intersects the y-axis at 2.

Example 2: x – 4y = 6

Step 1:

Example of slope-intercept form

Step 2:

Example of slope-intercept form

Step 3:

Example of slope-intercept form

In this case, the result is a descending line that intersects the y-axis at -1.5.

References