STEEMIT MATH CLASS | LESSON 2 - LINEAR EQUATIONS.

Hello Everyone, you are welcome to Steemit Math Class with @whileponderin. In the last class, we introduced a branch of mathematics known as Algebra. We said it is the branch of mathematics which represents every mathematical problem as a mathematical expression or equation. We also listed out the branches of Algebra, rules and properties of algebra as well as several area in the mathematics where Algebra can be applied.

_{Created using Canvas}

Just as promised, in today's class, we will be discussing about Linear Equations. If you haven't read the last class, you can read about it here Lesson 1: Algebra

LINEAR EQUATIONS

A linear equation can be defined as an algebraic equation (meaning that it has variable(s)) in which the highest index or power of the variable(s) is 1. What this means is that, none of the variables present in the equation should be raised to a power greater than one. Another name for a linear equation is a one-degree equation.

A linear equation can have more than one variable, but none of them should have a power greater than one. For a linear equation that has only one variable, it is always represented in this format:

Ax + B = 0.

In this equation above, "x" is the variable, while "A" is the coefficient, and "B" is the constant in the equation.

Examples of Linear equations with one variable includes

2x + 5 = 11

8b +6 = 9.

For a linear equation that has two variables it is always represented in this format:

Ax + By = C.

In this equation above, x and y are the variables, while A and B the coefficient, and C is the constant in the equation.

Examples of Linear equations with one variable includes

2x + 5c = 13

8b + 6d = 12.

Note that: For an equation to be said to have two variable, the letters representing the two variables must be different.

For example:

2x + 5c = 13 is a linear equation (as it's highest power or index is 1) with two variables "x" and "c", but

4x + 2x = 13 is a linear equation with only one variable, as both the 4x and the 2x can be added together to get 6x.

ADDITION AND SUBTRACTION OF ALGEBRA

Before we continue with linear equations, it's important that we learn how to add and subtract in algebra. Its knowledge will be applied in the solving of linear equations.

Below are the Rules for adding and subtracting Algebras.

1. Ax + Bx = (A + B)x.

This is because they both have the same variable (in this "x" case), so you add up the coefficients (in this case "A and B").

2. Ax - Bx = (A - B)x.

This is because they both have the same variable (in this "x" case), so you substract up the coefficients (in this case "A and B").

3. Ax + By = Ax + By.

This is because they have different variables, therefore you can't add their coefficients (even if the coefficients are the same).

4. Ax - By = Ax - By.

This is because they have different variables, therefore you can't substract their coefficients (even if the coefficients are the same).

Examples:

1. 2x + 4x = (2+4)x = 6x.

2. 2x - 4x = (2-4)x = -2x.

3. 3x + 4r = 3x + 4r.

4. 3x - 4r = 3x - 4r.

5. 4x + 4r = 4x + 4r

6. 4x - 4r = 4x - 4r

From these, you can see that in any algebraic equation (including linear equation) you need to pay more attention to the variable or variables present.

SOLVING LINEAR EQUATIONS WITH ONE VARIABLE

Linear equations of one Variable are easy to solve. The first thing you need to do is to separate the variable from the constant.The constants should be on one side of the equation, while the variable should be on the other side of the equation. Below are some examples to illustrate this

Example: Solve the linear equation with one variable: 2x + 7 = 21.

Solution:-
To solve this, I will first Identify the variable. The variable is "x", and it has a coefficient 2. The constants in the equation are 21 and 7. So, I will seperate both of them.

2x + 7 = 21

becomes

2x = 21 - 7 ( - sign changes for crossing equality sign)

Thus, 2x = 14.

To find x, we divide 14 by the coefficient of x.

That is:

x = (14/2)

x = 7.

Example 2:. 3x + 4 = 10.

Following the same procedure,

3x = 10 - 4,

3x = 6.

Therefore,

x = 6/3 = 2.

SOLVING LINEAR EQUATIONS WITH TWO VARIABLE

Linear equations containing two variables cannot be solved alone or individually. They can only be solved alongside another linear equation containing the same sets of variables. For instance, a linear equation 2x + 4y = 6, cannot be solved alone, as there is no way accurate way to solve for its variables. But when another linear equation containing the same x and y variables such as 3x - 6y = 12 is brought together with the first linear equation, both of them can be solved so as to obtain the true values of "x" and "y".

Some of the methods used in solving linear equations with two variables includes: substitution method, graphical method, the cross multiplication method, the determinant method, and the elimination method. We will discuss all these methods in our next class.

LINEAR EQUATION GRAPH

All linear equations can be illustrated using a graph. I'm sure you have seen a graph book before. A graph book is made up of lots of vertical and horizontal lines. Those vertical lines are called Y-axis or ordinate, while those horizontal lines are the X-axis or abscissa.

When a linear equation with only one variable is plotted on a graph, If the variable is "x" (or indicating a horizontal axis), then, the graph for the linear equation when plotted will be parallel to the y-axis ( or vertical axis). On the other hand, If the variable is "y" (or indicating the vertical axis), then, the graph for the linear equation when plotted will be parallel to the x-axis ( or horizontal axis).

Example 1: Plot a graph for a linear equation with one variable:

a. 2x + 4 = 20.

Solution:-

To plot this linear equation with one variable on a graph, follow these step:

Step 1: Make sure it's a linear equation.

Step 2: Note the variable in the equation and decide if should be on the vertical or the horizontal axis.

Step 3: Calculate the value of the variable from the equation.

Step 4: Since there is no other variable, assume that the other plotting point for the other variable is from - infinity to zero to infinity.

Step 5: Plot these points on your already scaled graph.

Based on these rules given, we know that the variable is "x", and should be on the horizontal axis. We can calculate the plotting point for x by making x the subject of formula.

Therefore, given that:

2x + 4 = 20

making x subject formula, we have

2x = 20 - 4. (The change in the operational sign is because the + crossed an equality sign, therefore turning to -)

2x = 16.

x = 16/2 = 8.

Therefore plotting point for our x-axis is 8. Since there is no other variable, we assume that the other axis (the vertical) axis should be from - infinity to 0 to infinity.

Below is the graph for the equation 2x + 4 = 20.

_{Linear Graph for the one-variable Linear Equation}

From this, you can see that the graph was plotted at the point 8 on the x-axis, according to scale. Then a straight line is drawn from the beginning of that graph to the end of the graph. Indicating that the value of the other axis can be any value from - infinity to 0 to infinity.

Example 2:- Plot a graph for a linear equation with two variables:

a. 2x + y = 20.

To plot this linear equation with one variable on a graph, follow these step:

Step 1: The linear equation is 2x + y= 20. It has variables x and y.

Step 2: Make the equation to be in this form: y = mx + b. If we do this, our equation becomes : y = 20 - 2x.

Step 3: Now, we can find plot this equation on a graph by picking random numbers of our choice as values for "x" and then find the corresponding value for y using the equation obtained from step 2.

Step 4: Assuming the values we picked for "x" are 0, 1, 2, and 3, when we substitute them individually into the equation obtained in step 2, we will obtain the following values for y

Y(0) = 20 - 2(0) = 20.

Y(1) = 20 - 2(1) = 18.

Y(2) = 20 - 2(2) = 16.

Y(3) = 20 - 2(3) = 14.

These results can be represented in the table below.

X	0	1	2	3
Y	20	18	16	14

Step 5: After we have obtained the values for y, we plot these points (0,20), (1,18), (2,16) and (3, 14) on a graph after which we will join them using a ruler to obtain a straight line graph as shown below. Note that these graph point are of the order (x, y) For instance, in (1, 18), 1 should be the cordinate of the point on the x-axis and 18 should be the cordinate of the same point on the y-axis.

The graph of the equation 2x + y= 20 is shown below:

_{Linear Graph for the two-variable Linear Equation}

You can use this online graph in plotting your linear graph. DESMOS GRAPHING

SUMMARY

In this post, we have been able, to explain what a linear equation is, as well as it's properties. We also talked about how to add and subtract algebra, how to solve linear equations with one variable and also how to plot a linear equation with one or two variables in a graph. In the coming sub-lessons, we will be discussing how to solve linear equations with two variables using the following methods: substitution method, graphical method, the cross multiplication method, the determinant method, and the elimination method. So, stay tuned to this blog. To check your progress for this lesson, try out these activities.

ACTIVITY

1. What is a linear equation? Give two examples each of a one variable and two variable linear equation.

2. Solve the following linear equations:

a. 2x + 7 = 11.
b. 5x - 16 = 24.

3. Classify the following into one- variable linear equation and two variable linear equation.

a. 5x + 2x = 13

b. 16c + 12b = 20

c. 18x + 9y = 0

d. 19 + 2c = 12.

4. Is √x + 3 = 12 a linear equation?

INTERESTING SCIENCE FACT: Did You know that ...

There are more bacteria in your mouth than there are people in the whole world!

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I hope you found this lesson informative. Till next time

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