A Review of Linear Equations and Inequalities (Part 1) PDF

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A Review of Linear Equations and Inequalities (Part 1)...

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Equations are two expressions connected by the equality symbol. I am using the term expressions because it is not only limited to numbers. Expressions can also be mathematical variables like x and y, or u and i. Or it could be anything. A few examples include:   

1+1=2 u + i = 69 Management Science = fun

Inequalities, on the other hand, are like equations, except that they use inequality symbols. Here are all the inequality symbols:   

≠ “not equal to” > “greater than” < “less than”

 

≤ “greater than or equal to” ≥ “less than or equal to”

The first inequality symbol is a vague one. It just refers to any other value not equal to the expression. For example, in x ≠ 17, possible values of x are every other number that is not 17. That’s a lot of possible values. For our succeeding lessons, we are not going to use this inequality symbol, although it is possible to plot such an inequality in the Cartesian plane.

Yes, speaking of the Cartesian plane, it will be heavily used in this lesson, and some other lessons going forward. That is why we are having this refresher lesson to properly refresh us for the next lessons.

Equations and Inequalities are quite vast. We have Quadratic equations and inequalities, Radical, Exponential, Logarithmic, and others. However, inasmuch as it would be fun to go back to those lessons that gave us our high grades and those Metrobank-MTAP-DepEd medals, we are not going to do that simply because we don’t need that in our future lessons. We are only going to refresh ourselves with Linear Equations and Inequalities.

Linear Equations and Inequalities are those composed with terms whose variables have exponents of 1 or below. For us to remember the parts of a linear equation, let us have the following example:

The entire thing is an equation, which, as mentioned previously is composed of two expressions connected by the equality symbol. This symbol is to be considered the “center” of any equation. There are two sides to any equation, one to the left of the equality symbol, and the other on its right. Each side of the equation is an expression. 8x + 3y - 12 is one expression. 60 is another expression. Each expression can contain one or more terms, separated by arithmetic operators. The most common arithmetic operators used are plus and minus. In the expression on the left, there are three terms: 8x, 3y, and 12 (or -12), a constant. To the right, there is only one term, 60, which is also a constant. A constant is a term with no variable. Actually, constants have variables, except that the exponent of these variables are zero. We know that anything raised to the power of zero is equal to 1. The terms 8x and 3y have numerical coefficients (8 and 3, respectively) and variables (x and y, respectively). The exponents are not shown, because they are understood to be equal to 1, if a variable is present. The exponent of x is 1. The exponent of y is also 1. 12 has a variable too, but its exponent is zero. As mentioned above, linear equations and inequalities are those using variables having 1 or below as exponents.

Next that we will be refreshing ourselves with are the properties of equality. At this point, we are going to concentrate on equations, since inequalities have slightly different properties. The following are the properties of equality: Property

Example

Reflexive

x=x

Symmetric

If x = y then y = x.

Transitive

If x = y and y = z then x = z

Addition

If x = y then x +z = y + z

Subtraction

If x = y then x - z = y - z

Multiplication

If x = y then xz = yz

Division

If x = y then x/z = y/z

Substitution

If x = y then y can replace x in any expression.

These properties are the bases of some techniques we use in solving systems of linear equations.

Next subtopic are the forms of linear equations. The equation of a line can be written in different forms, which depends on what is needed in plotting these lines in the Cartesian plane. The different forms are: Name Standard form Slopeintercept form Point-slope form

Form

Some notes

ax + by = c

y = mx + b



m = slope

 

b = y-intercept used as cost equation m = slope

 y – y1 = m(x - x1)

Two-point form

Intercept form

Vertical

x=a

Horizontal

y=b



(x1,y1) is a point in the line



(x1,y1) and (x2,y2) are points in the line

 

a = x-intercept b = y-intercept



Vertical line with a as x-intercept



Horizontal line with b as yintercept

Among these forms, we are going to encounter the standard form, the vertical line and horizontal line in our future lessons. It’s best that we concentrate on these three forms.

Next, we need to know how to plot linear equations in the Cartesian plane. Each linear equation will be represented in the Cartesian plane as a line, which is the set of all the points that satisfy the linear equation. A basic concept in linear equations is that two points make a line, and therefore, that is all we need to find out. Using the standard form, ax + by = c, it would be easy

to find the x-intercept, and the y-intercept. Take note that the x-intercept is the point where a line intersects with the x-axis. The x-axis is the set of points wherein the y-coordinate (also known as “ordinate”) is zero. Points such as (0,0), (1,0), (2,0), (3,0), and so forth, make up the x-axis. Conversely, the y-intercept is the point where a line intersects with the y-axis. The y-axis is the set of points wherein the x-coordinate (also known as “abscissa”) is zero. Points such as (0,0), (0,1), (0,2), (0,3), and so forth make up the y-axis.

To summarize, we only need to find two points:  

the x-intercept, which is obtained by setting y=0; and the y-intercept, which is obtained by setting x=0.

After which, we now have two points, and a line can be easily drawn connecting these two points.

To illustrate, let us use the linear equation used above: 8x + 3y - 12 = 60. Renaming the equation to follow the form ax + by = c, we get 8x + 3y = 72. The subtraction property of equality was used, wherein 4 was deducted from each side of the equation. Setting y to be equal to zero, we get:

Therefore, the line representing the linear equation above intersects the x-axis at the point (9,0).

To get the y-intercept, we set x to be equal to zero:

Therefore, the line intersects the y-axis at the point (0,24). Now we have two points: (9,0) and (0,24). Plot these points into the Cartesian plane, and connect these two points. The result would be the line representing the linear equation 8x + 3y – 12 = 60.

All the points in that line satisfy this linear equation. Let us try three points in this line to check whether they satisfy the equation.

Checking: Point #1 (3,16)

Point #2 (6,8)

Point #3 (7.5, 4)

Last thing we need to remember is how to solve systems of linear equations. Two or more linear equations with the same set of variables form a system of linear equations. The solution we obtain would be true for all equations in that system. To illustrate, we use the same linear equation, but we add a second linear equation: 6x – 4y = 4. The system of linear equations would then look like this:

There are four methods we can use to solve systems of linear equations: 



Graphical Method – useful if we need rough estimates, or if you’re sure that the answer is a full integer, without fractions or decimals. This simply requires plotting each of the equations in the system into the Cartesian plane. This was already exhaustively explained above. The point/s where the lines intersect is/are the solution/s. The best use of this method is to confirm the correctness of the solution obtained from the next three methods. Substitution Method – involves solving one of the equations for one variable in terms of the other. For this illustration, let us solve for x in terms of y using the second equation:

\

Since we now have a value of x in terms of y, we can now use the substitution property of equality, and substitute our x in the first equation with the value of x in terms of y:

In any of the equations, we can substitute y with 8 and the solve for x. For this illustration, we use the second equation:

Therefore, the solution to the system of linear equation above is (6,8). 

Linear Combination Method (or Elimination Method) – this involves adding one equation in the system to the other, or subtracting one equation from the other. The multiplication property of equality can be used to facilitate elimination. A variable must be eliminated thereafter, which makes is easier to solve for the remaining variable. After the value of the remaining variable is obtained, solve for the eliminated variable by substituting back the obtained value of the uneliminated variable into any of the equations. The following is how elimination method is done on the above system of equations:

The first equation is multiplied by 4, and the second equation is multiplied by 3 to make the coefficient of y in both equations equal to 12. This facilitates elimination. Since the signs of the coefficients of y are opposite, we just simply add both equations. If the signs of the coefficients are the same, subtraction is used. After obtaining the value of x, we substitute it with the value we obtained, which is 6....