Learn about Equations with Infinitely Many Solutions or No Solutions In this lesson, let's learn about equations that have infinitely many solutions or no solutions at all. What is an equation? An equation is basically two expressions that are separated by an equal to sign which means the value on the left is equal to the value on the right. Let's actually show an equation which has got many solutions. For example, on the left hand side, we've got x+4-6x. These are like terms, so let's combine them. We get 4 and then -6x+x is -5x, -6+1 is -5. Similarly, if I combine the right hand side, 6-2 is 4 and then I'm left with -5x. So, 4-5x on the left hand side and 4-5x on the right hand side, basically says no matter what I put as x, this will be true. So, this equation will be true for any value of x which means there could be infinitely many solutions. X could be 1, since they're both on the left and the right, they're exactly the same. I could replace x with any value and the equation would be true. This is true for any value of x. Let's try another one. Here let's try and combine like terms, -8x+9x is x+6=x-17. If I subtract x from both sides, what do I get? If I subtract x from both sides I get 0+6=-17. 6 is not equal to -17, so there's no value of x which will make this true. This equation has no solution because x+6 and x-17 can never be equal. How can you find the number which when you add 6 to it and subtract 17 from it will give you the same answer? This can never be true which means this equation is false or has no solution. Key thing to remember, if an equation would resolve it results in the same. Let me show you what I can do. If I subtract 4 from both sides, what do I get? -5x=-5x. Divide by both sides by -5, what do we get? x=x. This is true which means 4-5x and 4-5x is all true. What is the value for which x would be equal to x? Any value but here, when I solved for it, I get 6=-17 which is not true, so there is no solution.