Math 101 Lecture Notes 3.2 3.2 Graphing Linear Equations Ordered Pairs as Solutions to Equations
An ordered pair is a solution to an equation in two variables, x and y, if the given x–value
and y–value, when substituted into the equation, yield a true statement.
Example (a) Is (2, 3) a solution to y = 2x – 1?
(2, 3) is a solution to
y = 2x – 1 because
3 = 2 • 2 – 1 is a true statement
as shown at right. (2, 3)
y=2x –1
3=2•2–1?
3=4–1 ?
3 = 3 P Given an x–value or a y–value, we can complete
the ordered pair that satisfies an
equation in x and y by substituting the given value for the given variable and solving for
the other variable.
(0,
)
Example (b) For y = –3x + 4,
complete the ordered pair, (0,
).
To complete the ordered pair substitute 0
for x and solve for y as shown as right.
y = –3x + 4
y = –3 • 0 + 4
y=0+4
y = 4 The completed ordered pair is (0, 4)
Demonstration Problems 1. (a) Determine if (1, 5) is a solution to x – y = –4 2. (a) Complete the ordered pair (4, ) y = –3x + 5 Practice Problems 1. (b) Determine if (2, 1) is a solution to x + y = –3 2. (b) Complete the ordered pair (1, ) y = 2x – 4 Answers: 1. (b) No; 2. (b) (1, –2) Page 1 of 9 Math 101 Lecture Notes 3.2 Completing a Table of Values Completing a table of values is the same as completing a set of ordered pairs. Use the
method described on the previous page to complete the tables that follow.
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Demonstration Problems 1
3. (a) y = x −1 2
x y –2 –2 0 –1 2 0 4 1 4. (a) 2x – 3y = 6 Practice Problems 3. (b) y = 2x – 4 x y 4. (b) x + 4y = 4 x x y –3 –4 0 0 3 4 6 8 y Answers: 3. (b) –8, –6, –4, –2; 4. (b) 2, 1, 0, –1; Page 2 of 9 Math 101 Lecture Notes 3.2 Graphing a Linear Equation in Two Variables
A linear equation in two variables is one that can be written in the form
Ax + By = C
( A and B, not both 0)
We can describe the solution set of all ordered pairs of a linear equation by graphing the
equation. To graph a linear equation, we make a table of values of a few ordered pairs,
plot them, and then draw a line through them that extends across the coordinate plane.
Example (c) Graph 6x – 3y = 12
Step 1: Solve for y.
Step 2: Complete the table of values.
6x – 3y = 12
x
Step 3:
Step 4: Plot the ordered pairs from the
table of values and draw a line through the
points.
Largest x:
Smallest x:
Largest y:
Smallest y:
Draw axes that accommodate these values.
y
Page 3 of 9 Math 101 Lecture Notes 3.2 Using a Table of Values to Graph Equations that Contain Fractions
In making a table of values for a linear equation, we are free to choose any suitable x
values. If the coefficient of x is a fraction, it is best to choose x values that are multiples
of the denominator of the fraction.
Choose x-values that
are multiples of 3.
Example (d) Graph 2x – 3y = 6
Step 1: Solve for y.
Step 2: Complete the table of values.
2x – 3y = 6
x
–6
y
–3
0
3
6
Step 3:
Step 4: Plot the ordered pairs from the
table of values and draw a line through the
points.
Largest x:
Smallest x:
Largest y:
Smallest y:
Draw axes that accommodate these values.
Page 4 of 9 Math 101 Lecture Notes 3.2 Using X-­‐Intercepts and Y-­‐Intercepts to Graph Lines The x-­‐intercept is the point at which the line crosses the x-­‐axis. The y-­‐intercept is the point at which the line crosses the y-­‐axis. The x-­‐coordinate of the y-­‐intercept is 0. The y-­‐coordinate of the x-­‐intercept is 0. Example (e) Graph 6x – 3y = 12
Instead of completing a table of values of many ordered pairs, we can find the x-intercept
and y-intercept of the line, plot the two points and draw a line through them.
Step 1: Complete the table of values
(to find the x-intercept and y-intercept)
Step 3: Plot the two points from the table
of values and draw a line through them.
6x – 3y = 12
x
0
x-intercept
y-intercept
y
0
Step 2:
Largest x:
Smallest x:
Largest y:
Smallest y:
Draw axes that accommodate these values
adjusting the scale where necessary.
Page 5 of 9 Math 101 Lecture Notes 3.2 Graphing horizontal and vertical lines.
Example (f ) Graph y = 3.
Step 1: Write y = 3 as Step 3: Plot the values and draw a line through the points. 0x + y = 3 Step 2: Complete the table of values. x
–2
y
–1
0
1
2
Shortcut: Instead of completing a table of values, draw a line where y is always 3, that is, crossing (perpendicular to) the y axis at 3. Example (g) Graph x = –2.
Step 1: Write x = –2 as Step 3: Plot the values and draw a line through the points. x + 0y = –2 Step 2: Complete the table of values. x
y
–2
–1
0
1
2
Shortcut: Instead of completing a table of values, draw a line where x is always –2, that is, crossing (perpendicular to) the x axis at –2. Page 6 of 9 Math 101 Lecture Notes Demonstration Problems 5. (a) Find the x-­‐intercept and y-­‐intercept, then use them to graph 2x – 5y = 10 x
y
x-intercept
0
y-intercept
0 6. (a) Graph x = 6 3.2 Practice Problems 5. (b) Find the x-­‐intercept and y-­‐intercept, then use them to graph x + 4y = 4 x
y
x-intercept
0
y-intercept
0
6. (b) Graph y = –1 Answers: 5. (b) 4, 1; . 6. (b) . Page 7 of 9 Math 101 Lecture Notes 3.2 Interpreting a Graph as a Set of Ordered Pairs Given the graph of a linear equation, we can determine ordered pairs that are solutions to that equation by inspection. Example (h) Complete the ordered pairs using the graph at right. (i) (–4, ) 6 Find –4 on the x-­‐axis and find the 4 corresponding y-­‐value to the line. The completed ordered pair is (–4, 1) 2 (ii) ( , –3) –6 –4 –2 0 2 4 6 Find –3 on the y-­‐axis and find the corresponding x-­‐value to the line. –2 The completed order pair is (4, –3) –4 –6 Demonstration Problems Practice Problems Complete the ordered pairs using the Complete the ordered pairs using the graph above. graph above. 7. (a) (0, ) 7. (b) (2, ) 8. (a) ( , 2) 8. (b) ( , 0) Answers: 7. (b) –2; 8. (b) –2 Page 8 of 9 Math 101 Lecture Notes 3.2 Applications of Linear Equations Example (i) Create a graph that can be used to estimate the sales tax on a purchase under $100 made in Red Bluff. To find the Red Bluff sales tax, currently 7.5%, on a given amount, we can use the linear equation y = 0.075 x Complete the table of values and plot the ordered pairs. x (purchase amount) y (sales tax) 10 $ 0.75 $ 20 $ 1.50 $ 30 $ 40 $ 50 9 8 Sales Tax $ 10 6 5 4 3 Use the chart to estimate (i) the sales tax on a $45 purchase. (ii) the purchase amount if the sales tax is $5. (iii) the sales tax on a $90 purchase. 7 Page 9 of 9 2 1 10 20 3 0 40 50 60 7 0 80 90 100 Purchase Amount