Linear Equations in Two Variables

​LINEAR EQUATION IN TWO VARIABLES

Created by: SGPelaris/CRSHS​

​Illustrating Rectangular Coordinate System

A Rectangular Coordinate System or also known as a Cartesian plane is named after the French mathematician René Descartes (1596 – 1650),

who is known as the "Father of Modern Mathematics"

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​It is composed of two perpendicular number lines, typically called the x-axis and a the y-axis, respectively, that meet at point of origin (0, 0) and divide the plane into four regions called quadrants which are numbered in sequence as Quadrant I, Quadrant II, Quadrant III and Quadrant IV

​moving in a counter-clockwise direction starting from the upper right.

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Rectangular Coordinate System

Each point in the plane can be located using an ordered pair of numbers (𝑥, 𝑦),

where 𝑥 is the horizontal distance and 𝑦 is the vertical distance of the point from the origin. The numbers in the ordered pair are called coordinates.

The 𝑥 − 𝑣𝑎𝑙𝑢𝑒 of the coordinates (𝑥, 𝑦) of a point is also known as the abscissa,

while the 𝑦 − 𝑣𝑎𝑙𝑢𝑒 is known as the ordinate.

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Subject | Subject

An ordered pair (x,y) gives the coordinates of a point in a rectangular coordinate system.

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​The first number is the x-coordinate. The second number is the y-coordinate.

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The signs of the first and second coordinates of a point vary in the four

quadrants as indicated on the figure on the right.

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​The points which lie in

the 𝑥 − 𝑎𝑥𝑖𝑠 have coordinates (𝑥, 0) and the points which lie in the 𝑦 − 𝑎𝑥𝑖𝑠 have

coordinates (0, 𝑦), where 𝑥 and 𝑦 are real numbers.

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Remember: In naming the coordinates of a point, the first coordinate is the 𝑥 −

𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 and second is 𝑦 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒. They can’t be interchanged.

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Multiple Choice

A rectangular coordinate system is also known as ____________.

                                                   

                                                      

•   A.  Space coordinate system

• B    Polar coordinate system

• C.  Planar coordinate system

• D.  Cartesian coordinate system

Multiple Choice

How many quadrants does a Rectangular Coordinate System have?

                                                                                    

                                                                                   

• A.  2  

• B.  4    

• C.  6

• D.  8

Multiple Choice

Which of the following is true about the point (8, 1)?

• A.  It lies in Quadrant I.

• B. It lies in Quadrant II.

• C. It lies in both Quadrant II and III.

• D. Both statements A and C are correct

​Illustrating Linear Equations in Two Variables

An equation is said to be linear equation in two variables if it is written in the form of Ax + By + C = 0, where A, B & C are real numbers and the coefficients of x and y, i.e A and B respectively, are not equal to zero.

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​Examples:

10x + 4y = 3 and x + 5y = 2 are linear equations in two variables.

The solution of linear equations in two variables, Ax + By = C, is a particular point in the graph, such that when x-coordinate is multiplied by A and y-coordinate is multiplied by B, then the sum of these two values will be equal to C. 

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Basically, for linear equation in two variables, there are infinitely many solutions.

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Subject | Subject

Multiple Choice

If 𝐴, 𝐵, and 𝐶 are real numbers and if 𝐴 and 𝐵 are both not equal to 0, then

𝐴𝑥 + 𝐵𝑦 = 𝐶 is called a __________.

• A. Linear Equation in One Variable

• B. Literal Equation in Two Variables

• C. Linear Equation in Two Variables

• D. Linear Inequality

Multiple Choice

Which of the following is an example of Linear Equation in Two Variables

                                                                              

                                                                               

• A.  8x = 2+10  

•   B.  6x + 2y = 10

• C.  7x + 3y + 4z = 20

• D.  7a + 8b + 9c = 10+5

​Illustrating the Slope of A Line

​Rewriting Linear Equations in Two Variables

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Standard Form: 𝐴𝑥 + 𝐵𝑦 = 𝐶, where 𝐴, 𝐵 and 𝐶 ∈ ℜ, 𝐴 ≠ 0 and 𝐵 ≠

0; and

Slope-Intercept Form: 𝑦 = 𝑚𝑥 + 𝑏, where 𝑚 is the slope and 𝑏 is the

𝑦-intercept, 𝑚 and 𝑏 ∈ ℜ.​

​Rewriting Linear Equations in Two Variables

To write slope intercept form 𝑦 = 𝑚𝑥 + 𝑏 to standard form

𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0, let m = A/B, collect all terms on the left

side of the equation and multiply by the denominator B to

get rid of the fraction.

Multiple Choice

The steepness of the line also commonly known as rise over run is    called _______.

                                                                              

                                                          

• A. slope  

• B.  Linear Function

• C.  Linear Equation

• D.  Linear Inequality

Multiple Choice

The standard form of the equation of a line is 𝐴𝑥 + 𝐵𝑦 = 𝐶. Which of the following    equations is written in standard form?

                                                                                        

                                                                                

• A. 3𝑦 = 2𝑥 + 8     

•   B. 𝑥 + 5𝑦 = 6    

• C. −2𝑦 = 𝑥 + 2

• D. 2 + 𝑦 = 5

Multiple Choice

In his Math class, Bien Gregg  was asked by his teacher to rewrite  6𝑥 + 2𝑦 = 8 in slope-intercept form. He answered, 2𝑦 = −6𝑥 + 8. Was Bien Gregg correct?

• A. No, because the term −6𝑥 must be positive.

• B. Yes, because 6𝑥 and 8 is on the other side of the equation

• C. No, because the simplified answer must be 𝑦 = −3𝑥 + 4.

• D. Yes, because Bien Gregg is the smartest and never got wrong in the class.

Multiple Choice

𝐴𝑥+𝐵𝑦=𝐶 where A and B are not equal to zero is a linear equation in what form?

                                      

                           

• A. point- slope form

• B. slope-intercept form     

• C.  standard form

• D.  two-point form

​Graphing Linear Equations in Two Variables

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The graph of a linear equation is Ax + By = C is a straight line.

• Every point on the line is a solution of the equation.

• Every solution of this equation is a point on this line.​

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Multiple Choice

The following methods can be used in graphing a linear equation EXCEPT:

                                                           

                                                    

• A. slope    

• B. two points

• C. slope and a point

• D. 𝑥 and 𝑦- intercepts

Multiple Choice

Name the equation of the line graph given. 

• $$y\ =\ -\frac{2}{3}x\ -2$$  

• $$y\ =\ \frac{2}{3}x\ +2$$  

• $$y\ =\ \frac{3}{2}x\ -\ 2$$  

• $$y\ =-\ \frac{3}{2}x\ \ -\ 2$$  

Multiple Choice

What is the trend of the graph of a linear equation that has a slope of 5?

• A. The graph is a vertical line.

• B. The graph is a horizontal line.

• C. The graph is increasing from left to right.

• D. The graph is decreasing from left to right.

Multiple Choice

Hannah Noelle  was asked to describe the trend of the graph of an equation whose slope is 2. She answered, “it is increasing from left to right”. Is Hannah’s answer correct?

• A. Yes, because the graph is a slanting line.

• B. No, because the graph should be horizontal.

• C. No, because the graph should be a vertical line.

• D. Yes, because the equation whose slope is positive should be

                increasing from left to right.

1. Find the slope using the slope formula.

2. Find the y-intercept by substituting the slope and the coordinates of 1 point into the slope intercept formula, y = mx + b.

3. Write the equation using the slope and y-intercept.

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​Naming an Equation of a Line

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Multiple Choice

     𝐴𝑥+𝐵𝑦=𝐶 where A and B are not equal to zero is a linear equation in what form?

                                     

                             

• A. point- slope form

• B. slope-intercept form   

•   C.  standard form

• D.  two-point form

Multiple Choice

   What is the equation of a line that passes through the points (1,3) 𝑎𝑛𝑑 (−2,5)?

                                                             

                                                                                                          

• A. −2𝑥 + 3𝑦 = 11

•        B. 2𝑥 + 3𝑦 = 11

•   C. 2𝑥−3𝑦 = 11

• D. 3𝑥 + 2𝑦 = 11

Multiple Choice

  Given two points (𝑥₁,𝑦₁) and (𝑥₂,𝑦₂) where 𝑥₁≠𝑥₂, which of the following shall be used to determine the equation of the line?

• $$A.\ \ y\ =\ mx\ +b$$  

• $$B.\ \ y\ -\ y_1=m\left(x-\ \ x_1\right)$$  

• $$C.\ \ \frac{x}{a}+\frac{y}{b}=\ 1$$  

• D. $$y\ -y_1=\ \left(\frac{y_1-y_2}{x_1-x_2}\right)\left(x-x_1\right)$$  

Multiple Choice

Jonathan is six years older than Joshua. The sum of their ages is 48. Find Joshua’s and Jonathan’s age. What are the steps in solving Joshua and Jonathan’s age in chronological order?

I. Translate the problem into an equation, diagram, or chart.

II. Understand the problem. List all the components and data that are involved.

III. Check to see if the information is used correctly and that the answer makes   sense.

IV. Solve the equations you translated based on the given problem.

• A. I, II, III, IV                                                      

                                                                      

• C..  II, I, III, IV

•   B. II, IV, I, III

• D. II, I, IV, III

Multiple Choice

Mang Ruel earns P400 per day and additional P100 for every hour overtime as technician. What equation shall best represent the earnings of Mang Ruel when he rendered overtime services as a technician?

• A. 𝑦=𝑥 + 400                                                                      

                                                                                             

• B. 𝑦=2(𝑥+400)

• C. 𝑦=400𝑥 + 100

•     D. 𝑦=100𝑥 + 400

Multiple Choice

Based on the previous situation,  what is the amount earned by Mang Ruel when he rendered 3 hours of overtime?

                                                                           

                       

• A. P400     

• B. P500

• C. 700

• D. 800