LINEAR EQUATIONS - A Guide of Hints, Strategies and Simple Explanations - Algebra in Words

Algebra in Words: A Guide of Hints, Strategies and Simple Explanations (2014)

LINEAR EQUATIONS

A linear equation is an equation of the first degree; it produces a straight line. Lines are generally known to have:

·   a slope (m),

·   one y-intercept (b),

·   one x-intercept (there is no symbol, but the x-intercept is x when y = 0), and

·   the (slope-intercept) form: y = mx + b.

·   It could also be in standard form.

However, there are circumstances in which they will not have all of these criteria. I will summarize these and the three types of straight lines next.

A Diagonal Line:

A diagonal line will have a y-intercept, an x-intercept, and a slope of anything other than zero (or undefined). It will be in the form of

y = mx + b (when it is in slope-intercept form).

A Horizontal Line:

A horizontal line will be in the form:

“y = a number.”

It will have a y-intercept, and more specifically, the equation will be:

y = the y-intercept.

For instance, if the y-intercept is 3, the equation of the line will be

“y = 3”.

Students also mistakenly think the equation for a horizontal line will be in the form

“x =” (I think) because they associate the “x-axis,” with “horizontal.” But the opposite is the case, as explained above.

It is worth noting here that the equation for the x-axis is “y = 0” because it intersects the y-axis at “y = 0”.

Also, a horizontal line will have a slope of zero and will not have an x-intercept.

Students often mistakenly say horizontal lines have “no slope” (because the slope is zero), but this is incorrect. “No slope” does not mean “zero”. For more on this, see: The Slope Equation, and When y1 = y2.

A Vertical Line:

A vertical line is in the form of

“x = a number.”

A vertical line will have neither a y-intercept nor a slope, but it will have an x-intercept.

More specifically, the equation will appear in the form:

x = the x-intercept.

In other words, if the line intersects the x-axis at -5, then the equation for the line will be “x = -5”.

Students sometimes mistakenly think the equation of a vertical line will be in the form

“y =” (I think) because they associate the y-axis with being vertical.

It is worth noting here that the equation of the y-axis is “x = 0” because it intersects the x-axis at “x = 0”.

Another common mistake is to use “zero” and “no-slope” interchangeably, but they are significantly different. A vertical line literally has no slope (not even zero). It can also be said that the slope of a vertical line is “undefined.” For more on this, see: When x1 = x2, and: What Does "Undefined" Mean?

The concept of horizontal and vertical lines (and their equations) is something that students often have trouble with at first, perhaps because the books seem to give them a small section displaced from the more emphasized diagonal lines (which is just inevitable). Nevertheless, a good way to gain a stronger understanding of horizontal and vertical lines is to graph them (that way you can see them, and we all know what horizontal or vertical look like), and to have a good understanding of calculating slope, as shown in: The Slope Equation.

What Does “Undefined” Mean?

There are a variety of circumstances where “undefined” is used to describe the outcome of an equation. “Undefined” can sometimes be used in a similar context as “No Solution,” such as when computing an operation that can’t be done, like dividing by zero. Sometimes “Undefined” is used for times that a computation can’t be done, but isn’t referring to the answer of a problem. This could be the case when looking at the slope of a vertical line. The slope isn’t the “solution,” so “No Solution” isn’t appropriate… you would say the slope is undefined, or has “no slope.”

The most important thing about understanding “Undefined” is not using it synonymously with “Zero.” Also, when you do a computation on a calculator which would result as “Undefined,” your calculator will show “Error.”

How to Graph a Linear Equation

You can graph any equation… so don’t be afraid to do it at any time! Making a graph, whether you are asked to or not, is a great way to give clarity to your problem or answer, and is especially a great way to help you understand a problem or equation from a more visual and conceptual perspective. Graphing a linear equation is the easiest of all the types of possible equations.

To make a line, you need two or three or more points. Two points are the minimum number of points needed to make a line, but having a third point is better. Having a third point is a good check mechanism because if the three points do not fall into the same line (and instead, make a triangle), you know at least one of the points is wrong, and you must go back and correct it. If this is the case, I recommend starting your table of points over, since you won’t truly be able to tell which point (if not multiple points) is wrong. Also, the more points you have, the more accurate your line will be.

When dealing with linear equations, remember this:

When in doubt, make a graph.

To make a graph, make a table of 3 or more points.

Use the following procedure. First, draw the table, then fill in “0” for the first x, “0” for the second y, and “1” for the third x, as shown below.

x

y

0

0

1

Then take each number, substitute it into the original equation and solve for the other variable. This will give you three important points:

(x,y)

(0, ) ß the y-intercept, also known as b, or as a point (0, b)

( ,0) ß the x-intercept

(1, ) ß another easy point to find, near the origin

Sometimes, these points overlap, such as when the x-intercept and y-intercept are both at (0,0); or when the y-intercept is (1,0). That’s fine. Just make another point on the table. My next choice would be to put in “1” for y, then solve for x. You can really choose any starting number for either x or y, then substitute it in and solve to find its counterpart variable.

Here is another related piece of advice: If your slope is a whole number, write it over 1. For instance, if your slope is found to be m = 3, write it as x, because this will remind you that there is a rise and a run when you draw the graph.

If you have two equations (and their lines) to compare, be sure to make two separate tables so you can differentiate which points belong with which line. This could be useful for solving a system of two linear equations.

The Slope Equation

One major component of lines and graphing linear equations is the slope. The following shows all the interpretations of slope:

The symbolD is the capital Greek letter “D” which stands for “the change in,” commonly used in math and science equations.

The 4 Important Equations for Lines

These equations should be memorized, names included.

1. Slope-Intercept Form: y = mx + b

2. Standard Form: ax + by = c,

where a, b & c are #s, including possibly zero.

Note: the “b” here is not the same “b” (the y-intercept) as in the slope-intercept form. Although the same letter is used in each, they are used in completely different contexts.The letters “a” and “b” are typically used to represent coefficients in front of variables.

Also Note: A standard form linear equation is slightly different than a

Standard Form Quadratic Equation.

3) Slope: from the points:

(x1, y1) & (x2, y2)

4) Point-Slope Formula: y – y1 = m(x - x1)

Important comment about the Point-Slope Formula:

Keep y as y and x as x! Do not attempt to substitute values in for those here! You need them to remain (as letters) to the end of the process. The purpose of this formula is to substitute only values in for x1 & y1 (from a point) and the value of m, and then rearrange it into y = mx + b, where m and b will be numbers.

Look at the name… it’s the POINT-SLOPE formula… don’t overlook the name! You need one (x,y) point and the slope to substitute into it, which can be rearranged into

y = mx + b.

Sometimes, you will be given 2 or more points and no slope (m) and will be asked to find the equation of a line (as y = mx + b). In this case, you must first calculate m by using the two given points (or, if more than two are given, you must select any random set of two points) to put into the slope formula, and calculate m.

Next, choose one of the given points and put the corresponding values in for y1 and x1 and m (that you just determined) into the point-slope formula. Then, use the proper methods (rules of equality) to rearrange the point-slope formula into the slope-intercept formula, y = mx + b.

When x1=x2:

·   the slope is always undefined (and said to have “no slope”),

·   the line is vertical, and

·   the equation for the line will look like “x = #”.

Consider this example of the equation of a line going through the following points:

(4,3) and (4,7). Notice the x-values are the same, both 4, so in the equation:

the slope, m, is undefined because of the zero in the denominator. The equation of the line here is “x = 4”

When y1=y2:

·   the slope is always zero,

·   the line is horizontal, and

·   the equation of the line will look like “y = #”

Consider the following example of an equation of a line going through the two points (2,5) and (3,5). Notice the y-values are the same, both 5, and in the equation for slope:

since the numerator is zero, the slope, m, of the line is zero, and the

line is horizontal. The equation of this line would be “y = 5”

Parallel & Perpendicular Lines on a Graph

Lines are parallel when their slopes are identical.

In order to see this, you either need to:

·   Rearrange both equations into slope-intercept form and look at m, or

·   Simply calculate m for each equation and compare them.

·   You can also get a good idea by graphing and looking. If the lines cross, it will be fairly obvious.

It is not recommended to evaluate the slope (m) when equations are in standard form (or any form other than slope-intercept form). For more on this, see: No Solution - Inconsistent.

Lines are perpendicular when their slopes are exactly both opposite (and) reciprocals of each other. For example, if one slope is 4, the other must be . For more on this, see: One Solution - Consistent.