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.

 

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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

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, 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:

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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 real-world 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 work-related 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.


Example

Problem

Amanda’s dad is double as old together she is today. The amount of their eras is 66. Discover the ages of Amanda and also her dad.

We need to discover Amanda’s age and also her father’s age.

What is the trouble asking?

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Assign a variable to the unknown. The father’s period is 2 times Amanda’s age.

*

Amanda’s age added to she father’s period is equal to 66.

*

Solve the equation for the variable.

*

Use Amanda’s period to uncover her father’s age.

*

Do the answers make sense?

Answer

Amanda is 22 years old, and her dad is 44 year old.


Let’s try a new problem. Consider that the rental fee for a landscaping device includes a one-time 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.


Example

Problem

A landscaper wants to rental a tree stump grinder to prepare one area for a garden. The rental company charges a $26 one-time rental fees plus $48 because that each hour the machine is rented.

Write one expression for the rental price for any variety of hours.

The problem asks for an algebraic expression because that the rental expense of the stump grinder because that any variety of hours. An expression will have actually terms, one of which will contain a variable, yet it will not save an same sign.

What is the problem asking?

Look at the worths in the problem:

$26 = one-time fee

$48 = per-hour fee

Think about what this means, and shot to identify a pattern.

1 hr rental: $26 + $48

2 hr rental: $26 + $48 + $48

3 hr rental: $26 + $48 + $48 + $48

Notice the the variety of “+ $48” in the problem is the exact same as the variety of hours the maker is being rented. Since multiplication is repetitive addition, girlfriend could likewise represent it choose this:

1 hr rental: $26 + $48(1)

2 hr rental: $26 + $48(2)

3 hr rental: $26 + $48(3)

What information is essential to finding an answer?

Now let’s usage a variable, h, to represent the number of hours the machine is rented.

Rental for h hours: 26 + 48h

What is the variable?

What expression models this situation?

The full rental fees is identified by multiplying the variety of hours through $48 and including $26.

Answer

The rental price for h hrs is 26 + 48h.


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.


Example

Problem

A landscaper wants to rental a tree stump grinder come prepare one area because that a garden. The rental company charges a $26 one-time rental fee plus $48 for each hour the machine is rented.

What is the maximum number of hours the landscaper have the right to rent the tree stump grinder, if he deserve to spend no an ext than $290? (The maker cannot be rented for component of one hour.)

26 + 48h, wherein h = the variety of hours.

What expression models this situation?

*

Write an equation to help you discover out once the price equals $290.

Solve the equation.

*

Check the solution.

*

Interpret the answer.

Answer

The landscaper have the right to rent the maker for 5 hours.


It is often helpful to monitor a list of steps to organize and solve applications problems.

Solving application Problems

Follow these procedures to translate problem situations into algebraic equations you can solve.

1. Read and understand the problem.

2. Determine the constants and also variables in the problem.

3. Compose an equation to stand for the problem.

4. Settle the equation.

5. Inspect your answer.

6. Write a sentence that answers the concern in the applications problem.

Let’s try applying the problem-solving procedures with some brand-new examples.


Example

Problem

Gina has discovered a good price on record towels. She desires to stock up on these for her cleaning business. Paper towels cost $1.25 per package. If she has actually $60 to spend, how countless packages of document towels have the right to she purchase? create an equation that Gina might use to resolve this problem and show the solution.

The difficulty asks for how plenty of packages of record towels Gina deserve to purchase.

What is the difficulty asking you?

The record towels price $1.25 per package.

Gina has actually $60 to invest on paper towels.

What are the constants?

Let p = the variety of packages of paper towels.

What is the variable?

*

What equation to represent this situation?

Solve because that p.

Divide both sides of the equation by 1.25

60 ÷ 1.25 = 6,000 ÷ 125

5 00

1,000

1,000

0

*

Check her solution. Substitute 48 in for ns in her equation.

Answer

Gina can purchase 48 packages of document towels.


Example

Problem

Levon and Maria to be shopping for candles to decorate tables at a restaurant. Levon to buy 5 packages of candles plus 3 solitary candles. Maria bought 11 single candles to add 4 packages the candles. Every package the candles consists of the same number of candles. After ~ finishing shopping, Maria and also Levon realized the they had actually each purchased the same exact number of candles. How many candles room in a package?

The difficulty asks how numerous candles are included in one package.

What is the difficulty asking you?

Levon bought 5 packages and also 3 single candles.

Maria bought 4 packages and also 11 single candles.

What space the constants?

Let c = the variety of candles in one package.

What is the variable?

*

What expression represents the number of candles Levon purchased?

What expression represents the variety of candles Maria purchased?

What equation to represent the situation?

Maria and Levon purchase the same variety of candles.

*
 

Solve for c.

Subtract 4c indigenous both sides.

Subtract 3 indigenous both sides.

*

Check her solution.

Substitute 8 because that c in the initial equation.

Answer

There are 8 candle in one parcel of candles.


Advanced Example

Problem

The money from 2 vending equipments is gift collected. One maker contains 30 disagreement bills and also a bunch of dimes. The other an equipment contains 38 dissension bills and also a bunch the nickels. The number of coins in both makers is equal, and the amount of money that the machines gathered is likewise equal. How many coins room in every machine?

The trouble asks how countless coins room in every machine.

What is the problem asking you?

One maker has 30 dollar bills and also a bunch the dimes.

Another an equipment has 38 disagreement bills and also a bunch that nickels—the same variety of coins together the very first machine.

What room the constants and also what are the unknowns?

Let c = the number of coins in every machine.

What is the variable?

*

What expression to represent the amount of money in the an initial machine?

*

What expression to represent the quantity of money in the second machine?

*

What equation to represent the situation?

The amount of money in both machines is the same.

*

Solve because that c.

*

Check her solution.

Substitute 160 because that c in the initial equation.

Answer

There space 160 coins in each machine.


Advanced Question

Albert and Bryn room buying candy at the edge store. Albert buys 5 bags and also 3 separation, personal, instance pieces; Bryn buys 3 bags and then eats 2 pieces of candy from one of the bags. Each bag has the same number of pieces of candy.

After Bryn eats the 2 pieces, she has actually exactly fifty percent the variety of pieces the candy as Albert. How many pieces of candy room in each bag?

Pick the equation that might be supplied to fix the difficulty above. Use the change b to stand for the number of pieces of liquid in one bag.

A)

B)

C)

D)


Show/Hide Answer

A)

Incorrect. For the equation  to be correct, the case would have to be “Albert had actually 5 bags and 3 pieces, and Bryn had 3 bags and 2 pieces. Albert had fifty percent as much as Bryn.” The correct answer is: .

B)

Incorrect. Because that the equation  to it is in correct, the case would need to be “Albert had actually 5 bags and also 3 pieces, and Bryn had 3 bags and also 2 pieces. They had actually the very same amount.” The correct answer is: .

C)

Correct. The quantity of candy that Albert has can be represented by 5b + 3, and also the amount of liquid Bryn has actually can be stood for by 3b – 2. Due to the fact that Bryn has fifty percent as lot as Albert, the last equation is .

D)

Incorrect. Because that the equation  to be correct, the situation would have to be “Albert had actually 3 bags and 3 pieces, and also Bryn had 5 bags and then ate 2 pieces. Bryn had fifty percent as much as Albert.” The exactly answer is .

See more: A Difference Between Coelomates And Pseudocoelomates Is That


Summary


Some equations are taken into consideration special cases. These space equations that have no solution and also equations whose systems is the set of all genuine numbers. When you use the actions for resolving an equation, and you get a false statement quite than a value for the variable, over there is no solution. Once you usage the steps for resolving an equation, have avoided multiply both political parties of the equation by zero, and also you obtain a true statement rather than a worth for the variable, the equipment is all real numbers. Algebra is a an effective tool because that modeling and solving real-world problems.