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33

THE EQUATION OF A
STRAIGHT LINE

The equation of the first degree

Section 2

The x- and y-intercepts of a graph

y = 2x + 4

This is called an equation of the first degree.  It is called that because the highest exponent is 1.  

A solution to that equation is any values of x and y that will make the equation -- that statement -- true.  For example, the ordered pair (1, 6) is a solution.  Because upon replacing x with 1 and y with 6,

6 = 2· 1 + 4

6 is equal to 2· 1 + 4.

Another solution is (0, 4).  Because when  x = 0  and  y = 4, then

4 = 2· 0 + 4,

which again is true.

It's easy to find solutions.  Simply let x have any value you please -- the equation then determines the value of y.  For that reason, x is called the independent variable;  y is called the dependent variable.

Problem 1.   Find three solutions to the first degree equation  y = x + 4.

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For example:  (0, 4),  (1, 5),  (2, 6).

Problem 2.   Which of the following ordered pairs solve this equation:

y = 3x − 4 ?

(0, −4)   (1, 2)   (1, −1)   (2, −3)

(0, −4) and (1, −1). Because when x and y have those values, the equation is true.

The graph of a first degree equation

Consider the equation

y = 2x + 1.

Since there are two variables, x and y, then is it possible -- on the x-y plane -- to draw a "picture" of all the solutions to this equation?

First, to find a few solutions, complete this table.  That is, calculate the value of y that corresponds to each value of x:

x y = 2x + 1
0 1
1 3
2 5
−1 −1

Now plot these as points on the plane:

We see that all these solutions lie on a straight line.  In fact, the x, y coordinates of every point on that line will solve the equation!  Every coordinate pair is

(x, 2x + 1).

y = 2x + 1.

That line, therefore, is called the graph of the equation y = 2x + 1.  And  y = 2x + 1  is called the equation of that line.


Every first degree equation has as its graph a straight line.  (We prove that in Topics in Precalculus .)  For that reason, equations of the first degree -- where 1 is the highest exponent -- are called linear equations.

Problem 3.   

a)  An equation of the form  y = ax + b has what graph?

A straight line.  This is a linear equation.

b)  An equation of the form  Ax + By + C = 0 has what graph?

A straight line.  This is a linear equation. The capital letters are a convention for indicating integer coefficients.

Problem 4.   What characterizes a linear equation?

1 is the highest exponent.

Problem 5.   Which of the following are linear equations?

  a)   y = 4x − 5   b)   2x − 3y + 8 = 0   c)   y = x² − 2x + 1
 
  d)   3x + 1 = 0   e)   y = 6x + x3   f)   y = 2

a),  b),  d),  f).

Problem 6.

a)   Name the coordinates of any three points on the line whose equation
a)   is

y = 2x − 1.

(Pick any number for x; the equation will then determine the value
a)   of y.)

For example, (0 −1),  (1, 1),  (−1, −3).

Problem 7.

a)   Which of these ordered pairs solves the equation  y = 5x − 6 ?

(You have to test each pair!)

(1, −2)   (1, −1)   (2, 3)   (2, 4)

(1, −1) and (2, 4)

b)  Which of those are points on the graph of  y = 5x − 6 ?

(1, −1) and (2, 4)

Problem 8.   True or false?

a)  (−2, −3) is on the line whose equation is  x + y = 5.

False.

b)  (2, 3) is on the line whose equation is  x + y = 5.

True.

Constants versus variables

A constant is a symbol whose value does not change.   The symbols '5' and '' are constants.

The beginning letters of the alphabet a, b, c, etc. are typically used to denote constants, while the letters x, y, z , are typically used to denote variables.  For example, if we write

y = ax² + bx + c,

we mean that a, b, c are constants (i.e. fixed numbers), and that x and y are variables.

Problem 9.  The constants a and b.   Each of the following has the form  y = ax + b.  What number is a and what number is b?

  a)   y = 2x + 3.   a = 2,  b = 3. b)   y = x − 4.   a = 1,  b = −4.
 
  c)   y = −x + 1.   a = −1,  b = 1.   d)   y = 5x.   a = 5,  b = 0.
 
  e)   y = −2.   a = 0,  b = −2.   f)   y = −4x − 5.   a = −4,  b = −5.

Section 2

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