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?


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~ 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 real-world 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 work-related 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.


Example

Problem

Amanda’s dad is double as old together she is today. The sum of their periods is 66. Uncover the eras of Amanda and also her dad.

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

What is the problem asking?

*

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 same to 66.

*

Solve the equation because that the variable.

*

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

*

Do the answers do sense?

Answer

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


Let’s try a brand-new problem. Think about that the rental fee because that a landscaping maker includes a one-time 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.


Example

Problem

A landscaper desires to rent a tree stump grinder to prepare an area for a garden. The rental agency charges a $26 one-time rental fees plus $48 for each hour the maker is rented.

Write one expression because that the rental cost for any variety of hours.

The trouble asks because that an algebraic expression for the rental expense of the stump grinder for any variety of hours. One expression will have terms, among which will contain a variable, however it will certainly not contain an same sign.

What is the difficulty asking?

Look at the values in the problem:

$26 = one-time fee

$48 = per-hour fee

Think about what this means, and try to recognize a pattern.

1 hr rental: $26 + $48

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

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

Notice that the variety of “+ $48” in the difficulty is the same as the variety of hours the device is gift rented. Due to the fact that multiplication is repetitive addition, friend could likewise represent it prefer this:

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

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

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

What details is vital to detect an answer?

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

Rental for h hours: 26 + 48h

What is the variable?

What expression models this situation?

The total rental dues is figured out by multiply the variety of hours by $48 and including $26.

Answer

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


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 brand-new expression come solve another problem.


Example

Problem

A landscaper desires to rent a tree stump grinder come prepare an area because that a garden. The rental agency charges a $26 one-time rental fee plus $48 for each hour the an equipment is rented.

What is the maximum number of hours the landscaper have the right to rent the tree stump grinder, if he can spend no much more than $290? (The device cannot it is in rented for part of one hour.)

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

What expression models this situation?

*

Write one equation to assist you uncover out when the price equals $290.

Solve the equation.

*

Check the solution.

*

Interpret the answer.

Answer

The landscaper can rent the device for 5 hours.


It is often helpful to follow a perform of procedures to organize and also solve application problems.

Solving applications Problems

Follow these procedures to analyze problem cases into algebraic equations you deserve to solve.

1. Read and also understand the problem.

2. Determine the constants and variables in the problem.

3. Create an equation to represent the problem.

4. Settle the equation.

5. Check your answer.

6. Write a sentence the answers the inquiry in the application problem.

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


Example

Problem

Gina has discovered a an excellent price on document towels. She wants to share up on this for she cleaning business. Paper towels price $1.25 per package. If she has actually $60 to spend, how numerous packages of document towels deserve to she purchase? create an equation the Gina could use to fix this problem and show the solution.

The trouble asks for how plenty of packages of paper towels Gina have the right to purchase.

What is the problem asking you?

The paper towels expense $1.25 every package.

Gina has actually $60 to spend on document towels.

What are the constants?

Let ns = the variety of packages of document towels.

What is the variable?

*

What equation to represent this situation?

Solve for p.

Divide both political parties of the equation by 1.25

60 ÷ 1.25 = 6,000 ÷ 125

5 00

1,000

1,000

0

*

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

Answer

Gina can purchase 48 packages of document towels.


Example

Problem

Levon and Maria were shopping because that candles to decorate tables in ~ a restaurant. Levon purchase 5 packages of candles plus 3 single candles. Maria bought 11 single candles add to 4 packages the candles. Every package of candles includes the same number of candles. ~ finishing shopping, Maria and Levon realized that they had actually each purchase the very same exact number of candles. How many candles are in a package?

The trouble asks how plenty of candles are had in one package.

What is the difficulty asking you?

Levon bought 5 packages and also 3 solitary candles.

Maria bought 4 packages and also 11 single candles.

What space the constants?

Let c = the number of candles in one package.

What is the variable?

*

What expression to represent the number of candles Levon purchased?

What expression represents the variety of candles Maria purchased?

What equation represents the situation?

Maria and also Levon purchase the same number of candles.

*
 

Solve for c.

Subtract 4c native both sides.

Subtract 3 native both sides.

*

Check her solution.

Substitute 8 for c in the initial equation.

Answer

There are 8 candle in one parcel of candles.


Advanced Example

Problem

The money from 2 vending devices is gift collected. One maker contains 30 disagreement bills and also a bunch the dimes. The other an equipment contains 38 disagreement bills and also a bunch the nickels. The variety of coins in both makers is equal, and the quantity of money that the machines accumulated is also equal. How plenty of coins space in every machine?

The difficulty asks how plenty of coins space in every machine.

What is the trouble asking you?

One machine has 30 dissension bills and a bunch that dimes.

Another maker has 38 dollar bills and a bunch of nickels—the same variety of coins as the an initial machine.

What room the constants and what room the unknowns?

Let c = the number of coins in each machine.

What is the variable?

*

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

*

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

*

What equation represents the situation?

The quantity of money in both devices is the same.

*

Solve because that c.

*

Check her solution.

Substitute 160 for c in the original equation.

Answer

There are 160 coins in every machine.


Advanced Question

Albert and also Bryn room buying candy at the edge store. Albert buys 5 bags and also 3 individual pieces; Bryn buys 3 bags and then eats 2 pieces of candy from among the bags. Each bag has actually the same number of pieces that candy.

After Bryn eats the 2 pieces, she has exactly fifty percent the number of pieces of candy as Albert. How plenty of pieces the candy space in each bag?

Pick the equation that might be used to fix the problem above. Usage the change b to stand for the variety of pieces of candy in one bag.

A)

B)

C)

D)


Show/Hide Answer

A)

Incorrect. For the equation  to be correct, the instance would need 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 exactly answer is: .

B)

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

C)

Correct. The amount of candy the Albert has can be represented by 5b + 3, and also the amount of liquid Bryn has can be stood for by 3b – 2. Since Bryn has fifty percent as much as Albert, the last equation is .

D)

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

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Summary


Some equations are considered special cases. These are equations that have no solution and equations whose systems is the set of all actual numbers. When you use the measures for resolving an equation, and also you acquire a false statement quite than a value for the variable, over there is no solution. As soon as you usage the procedures for addressing 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 systems is all genuine numbers. Algebra is a an effective tool for modeling and solving real-world problems.