Hawkes Learning Systems: College Algebra

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Hawkes Learning Systems:College Algebra

Section 3.2: Linear Equations in Two Variables




Objectives

o Recognizing linear equations in two variables.o Intercepts of the coordinate axes.o Horizontal and vertical lines.




Recognizing Linear Equations in Two Variables

Linear Equations in Two VariablesA linear equation in two variables, say the variables and , is an equation that can be written in the form

where , , and are constants and and are not both zero. This form of such an equation is called the standard form.

xy

ax by c

a b c a b




Example 1: Linear Equations

Determine if the equation is a linear equation.

2 3 5 2 1x y x x y

2 6 5 2 1x y x x y

3 6 2 1x y x y

4 4 1x y

The equation is linear.

3 6 2 1x y x y





Determine if the equation is a linear equation. 4 3 5 7x x y x

4 3 15 3 7x x y x 7 7 3 15x x y

3 15y 5y


Note: One of the variables is absent from the resulting equation, but since the coefficient of is non-zero, this equation is still linear.

y





Determine if the equation is a linear equation.3

4 8x xy

34 4 8x xy

2 38 8 4x x y

3 38 4x y






Determine if the equation is a linear equation. 2 5 6 2 3 2x x y y x

2 5 6 6 4x x y y x 6 6 4 5x x y y

0 9The equation is not linear.

Note: The equation is not linear because the coefficients of and are both 0. Moreover, this equation has no solution: no values for and result in a true statement.

x yx y





Determine if the equations are linear equations. a. 36 3 7y x y

The equation is not linear.

Note: The presence of the cubed term in this already simplified equation makes it clearly not linear.

b. 224 2 3 8x x y 2 24 4 12 9 8x x x y

12 8 9x y





Intercepts of the Coordinate Axes

If the straight line whose points constitute the solution set crosses the horizontal and vertical axes in two distinct points, knowing the coordinates of these two points is sufficient to graph the complete solution.

x-axis

y-axis

1 1,x y

2 2,x y




Intercepts of the Coordinate Axes

o For an equation in the two variables x and y, it is natural to call the point where the graph crosses the x-axis the x-intercept, and the point where it crosses the y-axis the y-intercept.

o The y-coordinate of the x-intercept is 0, and the x -coordinate of the y-intercept is 0.

x-axis

y-axis

y-intercept

x-intercept




Example 6: Intercepts

Find the - and -intercepts of the equation and graph.x y3 4 12x y

3 0 4 12y 3y

3 4 0 12x 4x

-intercept: 0, 3y

-intercept: 4,0x




Example 7: Intercepts

Find the - and -intercepts of the equation and graph.x y 4 3 2 7x x y

5 2 10x y

5 0 2 10y

-intercept: 0,5y5y

5 2 0 10x 2x

-intercept: 2,0x




Horizontal and Vertical Lines

o A given linear equation may not have one of the two types of intercepts. This can only happen when the graph of the equation is a horizontal or vertical line.

o In the absence of other information, it is impossible to know if the solution of a linear equation missing one of the two variables consists of a point on the real number line, or a line in the Cartesian plane.

o You must rely on the context of the problem to know how many variables should be considered.




Horizontal and Vertical Lines

Consider an equation of the form . The variable is

absent, so any value for this variable will suffice, as long as

we pair it with . Thinking of the solution set as a set

of ordered pairs, the solution consists of ordered pairs with

a fixed first coordinate and arbitrary second coordinate.

This describes a vertical line with an -intercept of .

Similarly, the equation represents a horizontal line

with -intercept equal to .

ax c y

cxa

x ca

by c

y cb




Example 8: Horizontal and Vertical Lines

Graph the following equation. 3 2 7 2 5x x y x

5 5 14 2 0x x y 2 14y

7y

Note: The graph of the equation is the horizontal line consisting of all those ordered pairs whose y-coordinate is 7.





Graph the following equation. 5 0x 0x

Note: the graph of this equation is the -axis, as all the ordered pairs on the -axis have an -coordinate of 0.

yy x





Graph the following equation.2 2 3x

52

x

Note: This equation is a vertical line, which passes through 5/on the x-axis. 52

Hawkes Learning Systems: College Algebra

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