Page 451 - Algebra 1

L E S S O N Solving and Classifying Special Systems of
67
Linear Equations
Warm Up
New Concepts
1. Vocabulary A(n) of linear equations is a set of linear
(55)
Find the slope and y-intercept.
2. y = 4x - 7 3. 6x - 2y = 18
(49) (49)
4. Find the slope of a line parallel to 15x + 3y = 24. (65)
5. Find the slope of a line perpendicular to x + 4y = 7. (65)
Systems of linear equations can be classified by their common solutions. If no common solution exists, the system consists of inconsistent equations. The graphs of inconsistent equations never intersect. Therefore, since parallel lines never intersect, the graphs of inconsistent equations are the graphs of parallel lines.
equations with the same variables.
Example
1
Solve.
-3x + y = -4
y = 3x SOLUTION
Solving Inconsistent Systems of Equations
Caution
Be sure to isolate a variable before using it to solve by substitution.
Use substitution.
y = 3x - 4
y = 3x
3x = 3x - 4
Isolate y.
Substitute 3x for y in the first equation. Subtract 3x from both sides of the equation.
0 = -4
The statement is false. This means the system has no solution, so it is an
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inconsistent system.
Other systems of equations, known as dependent systems, can have an infinite number of solutions. The equations of a dependent system are called dependent equations, and they have identical solution sets. Since they have identical solution sets, the equations are the same.
Two methods can be used to solve dependent systems algebraically. One method shows that the equations are identical. The other method shows that the variables in dependent systems can be assigned any value. So, both equations have infinitely many solutions—an infinite set of ordered pairs.
436 Saxon Algebra 1