Combine Terms that contain the very same variables raised to the exact same powers. For example, 3x and −8x are favor terms, as are 8xy2 and also 0.5xy2.
You are watching: Which equation has the solution x 3
")">like terms on both sides of the equation.
Isolate the x hatchet by individually 2x indigenous both sides.
This is not a solution! girlfriend did not find a value for x. Fixing for x the method you recognize how, you arrive at the false statement 4 = 5. For sure 4 can not be equal to 5!
This might make sense as soon as you consider the second line in the solution where like terms were combined. If you multiply a number by 2 and add 4 you would certainly never acquire the same answer as as soon as you multiply that same number through 2 and add 5. Because there is no worth of x that will ever before make this a true statement, the equipment to the equation above is “no solution”.
Be mindful that you do not confuse the systems x = 0 v “no solution”. The equipment x = 0 way that the value 0 satisfies the equation, so over there is a solution. “No solution” method that over there is no value, not also 0, i m sorry would fulfill the equation.
Also, be cautious not to do the failure of thinking that the equation 4 = 5 way that 4 and 5 space values because that x that are solutions. If friend substitute these values into the original equation, you’ll see that they carry out not accomplish the equation. This is since there is important no solution—there are no worths for x that will make the equation 12 + 2x – 8 = 7x + 5 – 5x true.
Example  
Problem  Solve because that x. 3x + 8 = 3(x + 2)  
Apply the distributive residential property to simplify. Isolate the change term. Because you know that 8 = 6 is false, there is no solution.  
Answer  There is no solution.  
Advanced Example  
Problem  Solve for y. 8y = 2<3(y + 4) + y>  
Apply the distributive property to simplify. Once two to adjust of group symbols space used, advice the inner collection and climate evaluate the external set.  
 Isolate the change term by individually 8y indigenous both political parties of the equation. Due to the fact that you recognize that 0 = 24 is false, over there is no solution.  
Answer  There is no solution. 
Algebraic Equations v an Infinite variety of Solutions
You have seen that if an equation has no solution, you end up v a false statement rather of a value for x. You can probably guess the there might be a way you could end up with a true statement instead of a value for x.
Example  
Problem  Solve because that x. 5x + 3 – 4x = 3 + x  
 Combine prefer terms top top both sides of the equation. Isolate the x term by subtracting x from both sides. 
You arrive at the true explain “3 = 3”. Once you finish up with a true statement choose this, it means that the solution to the equation is “all real numbers”. Shot substituting x = 0 right into the original equation—you will acquire a true statement! shot
, and also it additionally will check!This equation wake up to have actually an infinite variety of solutions. Any value for x the you deserve to think of will certainly make this equation true. As soon as you think about the paper definition of the problem, this renders sense—the equation x + 3 = 3 + x way “some number add to 3 is same to 3 add to that very same number.” We know that this is always true—it’s the commutative residential property of addition!
Example  
Problem  Solve for x. 5(x – 7) + 42 = 3x + 7 + 2x  
Apply the distributive property and combine choose terms come simplify. Isolate the x hatchet by individually 5x from both sides. You gain the true statement 7 = 7, therefore you understand that x can be all actual numbers.  
Answer  x = all real numbers 
When fixing an equation, multiplying both sides of the equation by zero is not a an excellent choice. Multiply both next of an equation through 0 will certainly always an outcome in an equation that 0 = 0, but an equation of 0 = 0 does not assist you understand what the solution to the initial equation is.
Example  
Problem  Solve for x. x = x + 2  
Multiply both sides by zero. While that is true that 0 = 0, and you may be tempted to conclude that x is true the all genuine numbers, the is not the case.  
Check: Better Method:  For example, check and also see if x = 3 will deal with the equation. Clearly 3 never equals 5, so x = 3 is not a solution. The equation has no solutions. It was not beneficial to have multiplied both political parties of the equation by zero. It would have actually been better to have started by individually x from both sides, resulting in 0 = 2, leading to a false statement informing us that there room no solutions.  
Answer  There is no solution.  
In fixing the algebraic equation 2(x – 5) = 2x + 10, you finish up v −10 = 10. What does this mean? A) x = −10 and 10 B) over there is no equipment to the equation. C) friend must have actually made a failure in addressing the equation. D) x = all real numbers Show/Hide Answer A) x = −10 and 10 Incorrect. Any type of solution come an equation must satisfy the equation. If you substitute −10 into the original equation, you obtain −30 = −10. If you substitute 10 for x in the original equation, you acquire 10 = 30. The correct answer is: over there is no solution to the equation. B) there is no systems to the equation. Correct. Anytime you end up through a false statement like −10 = 10 it way there is no solution to the equation. C) friend must have made a mistake in resolving the equation. Incorrect. A false statement like this looks favor a mistake and also it’s always an excellent to inspect the answer. In this case, though, over there is no a failure in the algebra. The correct answer is: over there is no equipment to the equation. D) x = all genuine numbers Incorrect. If you instead of some real numbers into the equation, you will see that they execute not satisfy the equation. The correct answer is: over there is no equipment to the equation. 
Advanced Question How countless solutions space there for the equation: A) there is one solution. B) There are two solutions. C) There are an infinite variety of solutions. D) There room no solutions. Show/Hide Answer A) there is one solution. Incorrect. Shot substituting any value in because that y in this equation and think around what friend find. The correct answer is: There space an infinite number of solutions come the equation. B) There are two solutions. Incorrect. Shot substituting any kind of two worths in for y in this equation and think around what friend find. When taking care of sets the parentheses, make sure to advice the inner parentheses first, and also then relocate to the external set. The exactly answer is: There room an infinite variety of solutions to the equation. C) There are an infinite number of solutions. Correct. As soon as you evaluate the expressions on either side of the equates to sign, you acquire . If you to be to relocate the variables come the left side and the constants to the right, you would end up through 0 = 0. Since you have a true statement, the equation is true because that all values of y.D) There room no solutions. Incorrect. Recall the statements such together 3 = 5 are indicative of an equation having actually no solutions. The correct answer is: There are an infinite variety of solutions to the equation. Application Problems The strength of algebra is just how it can assist you version real situations in order to answer questions around them. This requires you to be able to translate realworld troubles into the language of algebra, and then be able to interpret the outcomes correctly. Let’s start by exploring a basic word trouble that provides algebra for its solution. Amanda’s dad is double as old as she is today. The sum of their eras is 66. Usage an algebraic equation to discover the periods of Amanda and her dad. One way to resolve this trouble is to usage trial and error—you deserve to pick part numbers for Amanda’s age, calculation her father’s age (which is double Amanda’s age), and then integrate them to check out if they workrelated in the equation. Because that example, if Amanda is 20, then she father would certainly be 40 because he is double as old as she is, however then their combined age is 60, no 66. What if she is 12? 15? 20? together you have the right to see, picking arbitrarily numbers is a really inefficient strategy! You can represent this situation algebraically, which provides another method to find the answer.
Let’s try a new problem. Consider that the rental fee for a landscaping device includes a onetime dues plus one hourly fee. You can use algebra to create an expression that helps you determine the full cost for a range of rental situations. One equation comprise this expression would certainly be helpful for trying to continue to be within a fixed price budget.
Using the information provided in the problem, you to be able to create a general expression because that this relationship. This means that you can uncover the rental expense of the machine for any number of hours! Let’s usage this new expression come solve one more problem.
It is often helpful to monitor a list of steps to organize and solve applications problems.
Let’s try applying the problemsolving procedures with some brandnew examples.
