Graphing Equations

Determine whether an ordered pair is a solution of an equation

An equation of the form 2x + 3 = 7 has one solution. An equation with one variable has only one solution. An equation with two variables, y = 3x + 2, has many solutions. For the equation y = 3x + 2, we write an ordered pair in the form (x, y). In any equation with two or more variables, we always consider the variables in alphabetical order. The order in which the variables appear in an equation does not change the ordered pair.

Example 1.

Determine whether (2, 4) is a solution of y = 2x - 4.

y = 2x - 4
4 = 2*2 - 4 Substituting 2 for x and 4 for y
4 = 0

The equation is false. The ordered pair (2, 4) is not a solution for y = 2x - 4.

Example 2

Determine whether (1, 1) is a solution to the equation 3 - 2x = y.

3 - 2x = y
3 - 2*1 = 1 substituting 1 for x and 1 for y.
1 = 1

The equation is true and therefore the ordered pair (1, 1) is a solution to the equation
3 - 2x = y.

Try Exercise #5

a. Determine whether (2, 3) is a solution of y = 5x + 3.
b. Determine whether (-2, 4) is a solution of 4y - 3x = 22.
Verify Your Answers Here!!!!!

We can find solutions of equations in two variables by choosing a value for one variable, substituting, and then solving for the value of the other variable.

Example 3

Find three solutions of y - 3x = -4

We choose any three values for x and then find the corresponding y values.

Our first value for x is 0. Thus, we have:

y - 3x = -4
y - 3*0 = -4
y - 0 = -4
y = -4

The ordered pair (0, -4) is a solution. Now choose the second x value.

y - 3x = -4
y - 3*1 = -4 substituting 1 for x
y - 3 = -4
y = -1

The second solution is (1, -1).
A third solution is (2, 2).

We can also record the solutions in a table format in order that it may be easier to read.

X Y

0 -4

1 -1

2 2

Try Exercise #6

Find three solutions of y - 3x = 1.
Use -2, -1, and 0 as your choices for x.
What are the three solutions you found?

Graphing Equations In Two Variables

We already know that the solution of an equation with two variables is an ordered pair of numbers, which can be plotted on a coordinate plane. A graph of an equation is a drawing that represents its solution. Using the three solutions you found to the equation y - 2x = -4, plot each coordinate. The points should all lie on a straight line. Thus we connect the dots to draw a line that will represent all solutions. Every point on the line is a solution to the equation y - 2x = -4.

Try Exercise #7

Graph the equation.

d. y - x = 1
Check Here For The Graph of The Equation.

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