VOCABULARY
Solution of an equation in two variables An ordered pair (x, y) that produces
a true statement when the values of x and y are substituted into the equation
Graph of an equation in two variables The set of points in a coordinate
plane that re present all solutions of the equation
linear equation An equation whose graph is a line or part of a line
Standard form of a linear equation Ax+By=5C, where A, B, and C are real
numbers and A and B are not both zero
Linear function The equation Ax+By=5C provided B ≠ 0 (that is, provided the
graph of the equation is not a vertical line).
Example 1 Check solutions of an equation
a. The ordered pair (-1, 2) is not a solution of x-2y=5 because
(-1)- 2(2)≠ 5.
b. The ordered pair (3, -1) is a solution of x-2y=5 because
(3)-2(-1)= 5.
Guided Practice Complete the fol lowing exercise .
1. Tell whether (-3, 5) is a solution of 6x +5y=8. Not a solution
Example 2 Graph an equation
Graph the equation x+y = 4.
Solution
Step 1 Solve the equation for y.
x + y = 4
y = 4 - x
Step 2 Make a table by choosing a few values for
x and finding
the values for y. |
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Step 3 Plot the points. Notice that the points
appear to lie on
a line.
Step 4 Connect the points by drawing a line through them.
Use arrows to indicate that the graph goes on without end.
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Guided Practice Graph the equation.
2. y = 2x - 1
3. y = -4x + 1
Example 3 De termine whether a point lies on a line
Tell whether the point lies on the line -x + 3y = 9.
a. (-2, 3) b. (6, 5)
Solution
a. The point (-2, 3) does not lie on the line -x + 3y = 9 because
-(-2) + 3(3) ≠ 9.
b. The point (6, 5) lies on the line -x + 3y = 9 because
-(6) + 3(5)= 9.
Example 4 Graph y = b and x = a
Graph (a)y = -3 and (b)x = 2.
Solution
a. For every value of x, the value of y is always -3 . The graph
of y = -3 is a horizontal line 3 units below the x-axis.
b. For every value of y, the value of x is always 2 . The graph of
the equation x = 2 is a vertical line 2 units to the right
of the y-axis.
Guided Practice Complete the following exercises.
4. Tell whether the point (-4, 2) lies on the line -x + 2y
= 8. Yes, the point lies on the line.
5. Tell whether the point (1, 25) lies on the line 3x - y
= -2. No, the point does not lie on the line.
6. Graph x = 0.5
7. Graph y = -1.5
EQUATIONS OF HORIZONTAL AND VERTICAL LINES
The graph of y = b is a horizontal line. The line y = b passes
through the point (0, b) .
The graph of x = a is a vertical line. The line x = a passes
through the point (a, 0) .
Example 5 Graph a linear function
Graph the function y = -x + 2 with domain x ≥ 0. Then
identify the range of the function.
Solution
Step 1 Make a table .
Step 2 Plot the points .
Step 3 Connect the points with a ray because the domain is
restricted .
Step 4 Identify the range. From the graph, you can see that all
points have a y-coordinate of 2 or more , so the range of the function is
y ≥ 2 .
Guided Practice Complete the following exercise.
8. Graph the function y = -x + 4 with domain x ≥ 0. Then identify
the range of the function.
range: y ≤ 4
