Combine Terms the contain the very same variables increased to the same powers. Because that example, 3x and −8x are favor terms, as are 8xy2 and 0.5xy2.
You are watching: Which equation has solution x = 3?
")">like terms ~ above both political parties of the equation.
Isolate the x ax by subtracting 2x from both sides.
This is no a solution! friend did not find a worth for x. Solving for x the method you recognize how, you come at the false explain 4 = 5. For sure 4 cannot be same to 5!
This might make sense once you think about the second line in the equipment where favor terms to be combined. If you main point a number through 2 and add 4 you would certainly never get the exact same answer as once you multiply that same number by 2 and add 5. Because there is no worth of x the will ever make this a true statement, the solution to the equation over is “no solution”.
Be careful that you perform not confused the equipment x = 0 with “no solution”. The systems x = 0 way that the value 0 satisfies the equation, so there is a solution. “No solution” way that there is no value, not even 0, i m sorry would satisfy the equation.
Also, be careful not to make the failure of reasoning that the equation 4 = 5 method that 4 and 5 room values for x that space solutions. If friend substitute this values into the original equation, you’ll see that they perform not accomplish the equation. This is because there is truly no solution—there room no values for x that will make the equation 12 + 2x – 8 = 7x + 5 – 5x true.
Example  
Problem  Solve for x. 3x + 8 = 3(x + 2)  
Apply the distributive property to simplify. Isolate the variable term. Due to the fact that you recognize 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 building to simplify. When two to adjust of group symbols room used, evaluate the inner set and then evaluate the external set.  
 Isolate the variable term by individually 8y from both political parties of the equation. Since you recognize that 0 = 24 is false, there is no solution.  
Answer  There is no solution. 
Algebraic Equations with an Infinite variety of Solutions
You have actually seen that if an equation has actually no solution, you finish up v a false statement instead of a worth for x. You can probably guess the there can be a means you might end up v a true statement rather of a worth for x.
Example  
Problem  Solve for x. 5x + 3 – 4x = 3 + x  
 Combine choose terms on both political parties of the equation. Isolate the x ax by individually x indigenous both sides. 
You come at the true statement “3 = 3”. As soon as you finish up v a true statement choose this, it means that the systems to the equation is “all actual numbers”. Shot substituting x = 0 into the original equation—you will obtain a true statement! shot
, and also it likewise will check!This equation happens to have actually an infinite number of solutions. Any kind of value for x that you have the right to think of will make this equation true. Once you think around the context of the problem, this makes sense—the equation x + 3 = 3 + x way “some number add to 3 is same to 3 to add that same number.” We understand that this is always true—it’s the commutative 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 ax by subtracting 5x native both sides. You obtain the true statement 7 = 7, therefore you understand that x have the right to be all real numbers.  
Answer  x = all genuine numbers 
When solving an equation, multiplying both sides of the equation through zero is no a an excellent choice. Multiply both next of an equation by 0 will always result in an equation that 0 = 0, yet an equation that 0 = 0 walk not assist you understand what the equipment to the original equation is.
Example  
Problem  Solve because that x. x = x + 2  
Multiply both sides by zero. While the is true that 0 = 0, and you may be tempted come conclude the x is true that all real numbers, that is no the case.  
Check: Better Method:  For example, check and also see if x = 3 will solve the equation. Clearly 3 never equates to 5, for this reason x = 3 is no a solution. The equation has actually no solutions. It to be not useful to have multiplied both political parties of the equation through zero. It would have actually been much better to have actually started by subtracting x indigenous both sides, resulting in 0 = 2, resulting in a false statement informing us that there space no solutions.  
Answer  There is no solution.  
In solving the algebraic equation 2(x – 5) = 2x + 10, you end up with −10 = 10. What does this mean? A) x = −10 and 10 B) there is no equipment to the equation. C) friend must have made a mistake in solving the equation. D) x = all real numbers Show/Hide Answer A) x = −10 and 10 Incorrect. Any type of solution to an equation must satisfy the equation. If you instead of −10 right into the original equation, you get −30 = −10. If you instead of 10 because that x in the initial equation, you get 10 = 30. The exactly answer is: there is no solution to the equation. B) over there is no solution to the equation. Correct. Whenever you finish up with a false statement prefer −10 = 10 it method there is no equipment to the equation. C) girlfriend must have made a mistake in fixing the equation. Incorrect. A false statement like this looks prefer a mistake and it’s always great to check the answer. In this case, though, over there is no a wrong in the algebra. The correct answer is: over there is no equipment to the equation. D) x = all actual numbers Incorrect. If you instead of some actual numbers into the equation, friend will check out that they do not meet the equation. The correct answer is: there is no systems to the equation. 
Advanced Question How numerous solutions space there because that the equation: A) there is one solution. B) There space two solutions. C) There room an infinite variety of solutions. D) There space no solutions. Show/Hide Answer A) there is one solution. Incorrect. Shot substituting any kind of value in for y in this equation and also think around what you find. The correct answer is: There space an infinite variety of solutions to the equation. B) There are two solutions. Incorrect. Shot substituting any kind of two values in for y in this equation and think around what friend find. When dealing with sets that parentheses, make certain to evaluate the within parentheses first, and then relocate to the outer set. The correct answer is: There space an infinite number of solutions come the equation. C) There are an infinite number of solutions. Correct. Once you advice the expression on either side of the equals sign, you acquire . If you to be to move the variables come the left side and also the constants to the right, you would finish up through 0 = 0. Since you have actually a true statement, the equation is true for all worths of y.D) There are no solutions. Incorrect. Recall that statements such together 3 = 5 space indicative of an equation having actually no solutions. The exactly answer is: There are an infinite number of solutions to the equation. Application Problems The power of algebra is just how it can help you version real cases in order to answer questions around them. This needs you to have the ability to translate realworld difficulties into the language that algebra, and then have the ability to interpret the outcomes correctly. Let’s begin by trying out a simple word trouble that provides algebra for its solution. Amanda’s dad is double as old as she is today. The sum of their ages is 66. Usage an algebraic equation to discover the eras of Amanda and her dad. One way to deal with this difficulty is to usage trial and also error—you deserve to pick some numbers for Amanda’s age, calculation her father’s age (which is double Amanda’s age), and also then combine them to check out if they workrelated in the equation. For example, if Amanda is 20, then she father would be 40 since he is double as old together she is, but then their merged age is 60, not 66. What if she is 12? 15? 20? together you can see, picking arbitrarily numbers is a very inefficient strategy! You can represent this situation algebraically, which provides another means to find the answer.
Let’s try a brandnew problem. Think about that the rental fee because that a landscaping maker includes a onetime dues plus one hourly fee. You can use algebra to produce an expression the helps you recognize the full cost for a variety of rental situations. One equation comprise this expression would certainly be valuable for do the efforts to remain within a fixed price budget.
Using the information provided in the problem, you to be able to develop a basic expression for this relationship. This method that girlfriend can discover the rental cost of the maker for any number of hours! Let’s use this brandnew expression come solve another problem.
It is often helpful to follow a perform of procedures to organize and also solve application problems.
Let’s shot applying the problemsolving procedures with some new examples.
