Two Equations with No Solutions
You saw above that a system of equations can have infinitely many solutions. When solving equations, you
likely assume, as most people do, that there will be at least one solution to the equation, but that is not
always the case. Look at the example below.
3 x − 6 = 3x + 7If we solve this equation we find that –6 = 7. Since –6 can never equal 7, there is no value of x that can be
put into this equation to make it true. In this case, the equation has no solutions.
What does it mean if two equations of lines have no solutions? Here’s one to try.
15.Which of the following accurately represents the set of solutions for the lines 6x + 12y = –24 and y = - x + 2 ?A) (0, –4)
B) (0, 4)
C) There are no solutions.
D) There are infinitely many solutions.There’s Just No Solution
When given two equations
with no solutions, find a
way to compare slopes.
The equations represent
parallel lines.Here’s How to Crack It
Start by putting the first line into y = mx + b form: 12y = –6x − 24. Divide the whole equation by 12, so y
= - x − 2. Since these lines have the same slope but different y-intercepts, the lines are parallel, and they
will never intersect. Therefore, (C) is the correct answer.
If two lines had different slopes, the lines would intersect at a single point such as (A) or (B). If the
equations were identical, then they would be the same line and therefore have infinitely many solutions.
Points of Intersection
In the chapters on algebra, we talked about how to find the solution to a system of equations. There are
several options, including stacking up the equations and adding or subtracting, setting them equal, or even
Plugging In the Answers. The SAT may also ask about the intersection of two graphs in the xy-plane,