Linear Equations in Two Variables June 26, 2012

Linear equations in two variables 2x + 3y = 6 y = 4x – 5 Note: In linear equation in two variables, The exponent of each variable must be 1 The two variables cannot be part of the term And no variable can appear in the denominator of a fraction

State whether or not the equation is a linear equation in two variables 1. 10x = 2y – 5 ____________________ 2. x2 + y2 = 1 ____________________ 3. 4xy + 2y =8 ____________________ 4. ____________________ 5. 6x – 4y + 3z = 12 ____________________ Not a Linear Equation in Two Variables Not a Linear Equation in Two Variables Linear Equation in Two Variables Not a Linear Equation in Two Variables

Equation of a line The most fundamental type of equation in two variables is a linear equation. A linear equation in two variables is an equation that can be written in the form. Ax + By = C This form of a linear equation in two variables is said to be in standard form. A standard graphing form of a linear equation in two variables is an equation that can be written in the form. y = mx + b in y-form (slope-intercept form).

Equation of a line Write each equation in standard form 1. 6x = 2y – 12 2. 6 = 7x – 2y 3. 4x + 3 = 3y + 5 4. 7x – 2y + 14 = 0 Write each equation in the y-form 1. x + 4y = 12 2. 3y = 6x + 2 3. 2x – 5y = 15 4. 3x = 8y

The sum of two numbers, x and y, is 8 x + y = 8 Based on the equation: If x = 2, y = x = 3, y = x = – 1 y = x = – 2 y = The ordered pairs (2,6), (3,5), (-1,9), (-2,10) are some of the solution of the equation x + y = 8. Thus, a solution of a linear equation in two variables is an _______________ that satisfies the equation. Can you give other solutions of this equation. How many solutions does the equation have? 6 5 9 10 ordered pair

Determine whether the pairs ( -2 , -10 ), ( 0 , 2 ) and ( 1 , 2 ) are solutions of the equation y = 4x – 2 Solution: To determine whether each pair is a solution, we replace x by first coordinate and y by second coordinate. When the replacements make the equation true, we say that the ordered pair is a solution. y = 4x – 2 –10 = 4(–2) – 2 –10 = –8 – 2 –10 = –10 true y = 4x – 2 2 = 4(0) – 2 2 = – 2 2 = –2 false y = 4x – 2 2 = 4(1) – 2 2 = 4 – 2 2 = 2 true ? ? ? ? ? ?

Determine whether or not the given ordered pair is a solution of the given equation. 1. 3x + 4y = 32 (4 , 5) ____________________ 2. y = 5x + 3 (-2 , 13) ____________________ 3. 2x = y + 5 (6 , 7) ____________________ 4. (3 , 4) ____________________ 5. 7x – 2y = 5 (1, – 1) ____________________ Not a Solution Solution Not a Solution Not a Solution

Complete the Ordered pairs for each equation 1. 5x + y = 12 (0 , __) ; (2 , __) ; (__ , 27) 2. x + y = 5 (__ , 5) ; (5 , __) ; (__ , -2) 3. x = 3y + 20 (5 , __) ; (__ , -2) ; (__ , 6) 4. y = –3x (2, __) ; (-2, __) ; (3, __) 5. x = 2y – 1 (__ , 3) ; (__ , ½) ; (__ , 0)

Graphing linear equations by point-plotting The basic method of graphing an equation is by point- plotting. The idea is to plot as many points that satisfy the equation, until a clear picture of the graph is drawn To graph the linear equations in two variables we follow the following steps: Change the equation into standard graphing form y = mx + b where m and b are real numbers Make a table of values at least three (assign a real number to x the independent variable then solve for the corresponding value of y the dependent variable) Plot the points on the Cartesian plane Draw a line connecting the point, then label it with the equation.

Complete the table then draw the graph 1. y = 3x – 5 . x 3 y 1 2 . -5 4 .

Complete the table then draw the graph 1. y = –4x + 2 x 1 y -6 2 . -2 2 . .

Complete the table then draw the graph 1. x – 3y = 7 x -2 4 y 1 -3 -1 . . .