THE EQUATION AND GRAPH
|x||y = 2x + 1|
We see that all those solutions lie on a straight line. In fact, every pair (x, y) that solves that equation will be the coördinates of a point on that line. On that line, every coördinate 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.
The graph of an equation, in other words, is the graph of its solutions. It is the picture of those values of (x, y) that make the equation -- that statement -- true.
Every first degree equation -- where 1 is the highest exponent -- has as its graph a straight line. (We prove that in Topics in Precalculus .)
For that reason, an equation of the first degree is called a linear equation.
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).
|a)||Name the coördinates of any three points on the line whose equation is|
y = 2x − 1.
|(Choose any number for x; the equation will then determine the value of y.)|
For example, (0 −1), (1, 1), (−1, −3).
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.
b) (2, 3) is on the line whose equation is x + y = 5.
Constants versus variables
A constant is a symbol whose value does not change. The symbols 2, 5, , are constants.
The beginning letters of the alphabet a, b, c, are typically used as arbitrary constants, to which numerical values may be assigned; 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 numbers, and that x and y are variables.
Problem 9. The arbitrary 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.|
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Copyright © 2021 Lawrence Spector
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