Solving linear equations and linear inequalities — Harder example
About this video
A linear equation has infinitely many solutions when it simplifies to a true statement for every x, such as x = x. The instructor simplifies 3 + 10x - 5 = (a + 1)(x - 2) and finds that a must equal 9, because then 9x = 9x holds for all x. The video also notes that other values of a force a single solution.
You'll be able to
- simplify both sides of a linear equation.
- identify when an equation has infinitely many solutions.
- find the value of a constant that makes an equation true for every x.
Key words
- Infinitely many solutions
- A situation where an equation is true for every value of the variable.
- Distributive property
- Multiplying a sum or difference by a number on each term inside, as in x(a + 1) = ax + x.
- Coefficient
- The number multiplying a variable, such as 9 in 9x.
- Linear equation
- An equation in which the variable appears to the first power only.
Transcript
- [Instructor] In the equation shown above, a is a constant. For what value of a does the equation have infinitely many solutions? So you end up with infinitely many solutions if your equation simplifies to something like x is equal to x, or one is equal to one, something that's true that's going to be true for any x that you pick.
So let's see what we could do with this thing right over here. These are obviously not, if you got 100 equals 100, that would be the same, that would have an infinitely many solutions. Zero equals zero. These were all be situations where you have an infinite number of solutions.
So when I look at this thing up here, my first instinct is, well let's just see if I can simplify this a little bit. I'll leave the a in there and then see if I can get to a point where it's gonna have an infinite number of solutions. So let me just rewrite it.
So we're gonna have three plus 10x minus five is equal to a plus one times x minus two. So let's see, on the left-hand here I can add the three and the negative five. Or I could take three minus five. That would be negative two. So I get 10x minus two is equal to, let me distribute the x.
So it's gonna be ax plus x. All I did here is I distributed the x, minus two. Now let's see, what happens if, let's see, I could get rid of both of these negative twos if I add two to both sides. So if I just, remember anything I do to one side I've gotta do to the other one if I wanna hold the equality to continue to be true.
So I just added two to both sides. And I'm left with 10x is equal to ax plus x. Let's see. Let's subtract x from both sides. So if I subtract x. Actually I could write it like this. I could subtract x from both sides. On the left-hand side I'm gonna get nine x.
On the right-hand side I'm gonna get ax. So how could I have an infinite number of solutions, an equation that's gonna be true for any x? Well if a was equal to nine, because if a is equal to nine I'm gonna have a situation. So if a is equal to nine then you're gonna have a situation where nine x is going to be equal to, instead of a, I'd write nine.
Is going to be equal to nine x. Well that's going to be true for any x. Any x times nine is going to be equal to that same x times nine again. You're gonna have an infinite number of solutions. And so a needs to be equal, a it needs to be equal to, a is equal to nine.
Now what's really interesting here is think about what would happen if a is any of these, if a is any of these other things right over here. Then you're going to force a different solution. But anyway, we'll leave that for another video.
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