Transcript Equations With Two Variables

PRE-ALGEBRA
Lesson 8-2 Warm-Up
PRE-ALGEBRA
Equations with Two Variables
(8-2)
What is a “solution”?
solution: an ordered pair that makes an equation with two variables a true
statement (in other words, the graph of the equation will pass through that
point)
Example: (1, 2) is a solution for y = 2x, because both sides of the equation
are equal (“true statement”) for x = 1, y = 2. [2 = 2(1) or 2 = 2 ]
How do you find a
solution when given
one of the two variables
of the equation?
To find the solution to an equation when given one of its two variables,
substitute the variable you know into the equation and solve for the other
variable.
Example: Solve y = 3x + 4 for x = -1.
y = 3x + 4
y = 3(-1) + 4
y = -3 + 4
y=1
Substitute x = -1 into the equation.
Simplify
Solve for y.
A solution for the equation y = 3x + 4 is (-1, 1).
PRE-ALGEBRA
Equations With Two Variables
LESSON 8-2
Additional Examples
Find the solution of y = 4x – 3 for x = 2.
y = 4x – 3
y = 4(2) – 3
y=8–3
y=5
Replace x with 2.
Multiply.
Subtract.
A solution of the equation is (2, 5).
PRE-ALGEBRA
Equations With Two Variables
LESSON 8-2
Additional Examples
The equation a = 5 + 3p gives the price for
admission to a park. In the equation, a is the admission
price for one car with p people in it. Find the price of
admission for a car with 4 people in it.
a = 5 + 3p
a = 5 + 3(4)
a = 5 + 12
a = 17
Replace p with 4.
Multiply.
Add.
A solution of the equation is (4, 17). The admission price for
one car with 4 people in it is $17.
PRE-ALGEBRA
Equations with Two Variables
(8-2)
What is a “linear
equation”?
linear equation: an equation in which every solution (ordered pair that makes
it a true statement) forms a line on a graph
Example: any equation in which the x is to the first power (i.e. not x 2, x3, x4,
etc.) will form a line when you graph its solutions
How can you use a
graph to fine the
solutions to a linear
equation?
To find the solutions to an equation using a graph: 1. create a function table
by making up your own x values to find the y values of at least three points on
the graph; 2. plot the points, and 3. draw a line through the points. Any
ordered pairs on the line you drew are also solutions to the equation.
Example: Graph y = - 1 x + 3. Is (2,2) a solution?
2
Step 1: Make a table of values to find at least three ordered pair solutions?
PRE-ALGEBRA
Equations with Two Variables
(8-2)
Step 2: Plot the ordered pair and draw a line through the points.
The point (2, 2) is on the line, so it is a solution to the equation.
Check: Substitute (2, 2) into the equation to make sure it makes a true
statement.
y=-1 x+3
2
2 = - 12 (2) + 3
Substitute 2 for x and 2 for y.
2 = -1 + 3
2=2
True statement!
PRE-ALGEBRA
Equations With Two Variables
LESSON 8-2
Additional Examples
Graph y = 4x – 2.
Make a table of values to show ordered-pair solutions.
x
–2
0
2
4x – 2
4(–2) – 2 = – 8 – 2 = –10
4(0) – 2 = 0 – 2 = –2
(x, y)
(–2, –10)
(0, –2)
4(2) – 2 = 8 – 2 = 6
(2, 6)
Graph the ordered pairs.
Draw a line through the points.
PRE-ALGEBRA
Equations with Two Variables
(8-2)
What if the graph of an
equation is a vertical or
horizontal line?
Example: Is y = 2 a function?
This is a horizontal line. In
other words, for every value
of x, y = 2. Since there is only
one x value for every y value,
the equation is a function.
Example: Is x = 2 a function?
This is a vertical line. In other
words, for every value of y, y
= 2. Since there are an
infinite number of points at x
= 2 (doesn’t pass the vertical
line test), the equation is
NOT a function.
PRE-ALGEBRA
Equations With Two Variables
LESSON 8-2
Additional Examples
Graph each equation. Is the equation a function?
a. y = –3
b. x = 4
For every value of x, y = –3.
For every value of y, x = 4.
This is a horizontal line.
The equation y = – 3 is
a function.
This is a vertical line.
The equation y = 4
is not a function.
PRE-ALGEBRA
Equations with Two Variables
(8-2)
How do you graph an
equation not in y =
form?
If the equation isn’t in y = form, you can solve for y before creating a function
table..
Example: Graph 3x + y = -5.
Step 1: Solve for y.
3x + y = -5
3x + y =
-5
- 3x
-3x
0 + y = -3x -5
y = -3x -5
Step 2: Make a table of
values to find at least three
ordered pair solutions.
Given
Subtract 3x from both sides.
Simplify.
Step 3: Graph the points
from your function table and
draw a line through them.
PRE-ALGEBRA
Equations With Two Variables
LESSON 8-2
Additional Examples
Solve y – 1 x = 3 for y. Then graph the equation.
2
Solve the equation for y.
y–
1
1
x=3
2
1
1
y – 2 x + 2 x = 3 +2
1
y=2 x+3
x
1
Add 2 x to each side.
Simplify.
PRE-ALGEBRA
Equations With Two Variables
LESSON 8-2
Additional Examples
(continued)
Make a table of values.
1
2
1
–2
2
1
0
2
1
2
2
x
Graph.
x+3
(x, y)
(–2) + 3 = –1 + 3 = 2
(–2, 2)
(0) + 3 = 0 + 3 = 3
(0, 3)
(2) + 3 = 1 + 3 = 4
(2, 4)
PRE-ALGEBRA
Equations With Two Variables
LESSON 8-2
Lesson Quiz
Find the solution for each equation for x = 2.
1. y = –2x + 5
2. y = 7x
(2, 1)
(2, 14)
3. y = 3x – 9
(2, –3)
Solve each equation for y. Then graph each equation.
4. y – 2x = 3
5. 2x + 2y = 8
y = 2x + 3
y = –x + 4
PRE-ALGEBRA