Solving Equations: Word Problems
Basics on the topic Solving Equations: Word Problems
Real world problems are not usually written as an algebra equation, so we must translate a math story into an equation.
How do we do that? By carefully analyzing the information that is presented in the form of a word problem. Look for keywords to indicate the operations you should use to write an equation. For example, five more than x can be written as x + 5. The statement, ten times the unknown number, is written as 10x. What is the meaning of two times the sum of 5 and x? The word sum tells us to put 5 and x inside parentheses, the result is 2 ( 5 + x ).
After you have written the equation to describe your math story, solve the equation by following the steps you have learned for solving algebraic equations. Remember if there are multi steps, combine like terms then use inverse operations to isolate the variable to calculate the solution.
It’s always a good idea to check your work by substituting the solution back into the equation you wrote then check the solution against the math story. If the solution works for the equation but not for the story, recheck your original equation.
Explain steps to solve algebraic equations. CCSS.MATH.CONTENT.HSA.REI.A.1
Transcript Solving Equations: Word Problems
We're at the secret hideout of Dr. Evil, located on Volcano Island. He needs some gold to buy a present for his mommy's birthday.
Mommy Evil likes bling, so she wants a ring with as many diamonds as possible. Every jewel on the ring cost a measely 10 gold bars (obviously the typical currency for villans). But the problem is: he's a bit short on gold.
Word Problem Example
To determine how much of his stuff he needs to sell, we should convert the information from the word problem into an equation.
First, he lists one of his private islands on Evil-Bay and his asking price is 50 gold bars - quite a bargain!
Next, he'll sell 20 of his most evil robotic rocket crocs for 5 gold bars each. Hopefully they sell quickly. There's not much time until Mommy Evil's birthday.
He wonders how much bling he can put on the ring if each diamond costs 10 gold bars each. Let the number of diamonds equal x.
Translating the Word Problem and Writing the Equation
Let's write an equation. On the left side of the equal sign, write the price of his private island times the number of islands, 50 times 1, and add the price times the number of rocket crocs he plans to sell, 5 times 20.
On the right side, write the cost in gold bars, 10, times the unknown amount, x, which stands for the total number of diamonds he can put on his mom's ring.
Solving the Word Problem Example 1
Now, to solve the equation for x, you have to simplify the equation using PEMDAS and opposite operations.
- Here, multiplication is first. 50 · 1 = 50 and 5 · 20 = 100.
- Now we have to add: 50 + 100 = 150.
- 150 = 10x. To isolate x, we use the opposite operation here. We divide both sides by 10.
- x = 15. The ring for Mommy could have 15 diamonds.
Solving the Word Problem Example 2
But it's Mommy's 60th birthday and Dr. Evil decides he must buy her a ring with 60 diamonds. Wow! In addition to the island, how many rocket crocs will he need to sell? Let this now be x.
Now let's write another equation. Again, write the price of the island multiplied by the number of islands for sale and add the price of the rocket crocs times x on the left side of the equation.
Make this equal to the price of the gold bars times the number of diamonds Dr. Evil wishes to put on his mom's ring. Simplify the equation using PEMDAS and solve for the unknown variable using opposite operations.
- Again we start with multiplication here: 50 · 1 = 50 and 10 · 60 = 600.
- Now let's get rid of the 50 to isolate x. We subtract 50 on both sides. So 5x = 550.
- Now isolate x by dividing both sides by 5.
- x = 110. Dr. Evil will need to sell 110 rocket crocs.
Solving the Word Problem Example 3
What? You've gotta be kidding! He has only 55 of the rockets crocs! Oh boy, what can he do? Eureka! Dr. Evil realizes he can raise the price of the rocket crocs. Back to the equation...
This time, the unknown number, 'x' is the new price of the rocket crocs. So, again, write the price of the island multiplied by the number of islands for sale and add the price of the rocket crocs, our new x, times 55, the number of rocket crocs he has, on the left side of the equation.
Set this equal to the value and the number of diamonds on the ring. Simplify and solve.
- Multiplication first: 50 · 1 = 50, x · 55 = 55x, 10 · 60 = 600.
- Now to get rid of this 50, we use opposite operations by subtracting 50 from both sides. So 55x = 550.
- To isolate x here we divide both sides by 55.
- x = 10. This means he must sell the rocket crocs for 10 gold bars each.
Great, now that he's finished with his calculations, he can start the sale. Oh no! One of the robotic rocket crocs just exploded! Maybe Dr. Evil can buy smaller diamonds?
Solving Equations: Word Problems exercise
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Examine the number of diamonds Dr. Evil can buy for his mom's ring.
HintsPEMDAS means the order of calculation
- Paranthesis
- Exponent
- Multiplication
- Division
- Addition
- Subtraction.
The opposite operation is given by
- $+~\longleftrightarrow~-$ and
- $\times~\longleftrightarrow~\div$.
The solution is a natural number.
SolutionFor his mom's birthday present, Dr. Evil would like to buy a ring. His mom loves diamonds, but one diamond costs $10$ gold bars. Dr. Evil considers selling one island for $50$ gold bars and $20$ rocket crocodiles for $5$ gold bars each.
After he sells the island and rocket crocs, how many diamonds can Dr. Evil buy for his mom?
The number of diamonds is unknown. So we assign the variable $x$ to it. We know the cost of one diamond is $10$ gold bars. So $x$ diamonds cost $10\times x$ gold bars.
On the other hand, Dr. Evil gets:
- $50\times 1$ gold bars for his island
- $5\times 20$ gold bars for $20$ rocket crocodiles
- All together, he has $50\times 1+5\times 20$ gold bars.
$50\times 1+5\times 20=10\times x$
We simplify this equation by using PEMDAS and then isolating $x$ by performing Opposite Operations. Let's do this step-by-step:
- First, we multiply: $50+100=10x$,
- Next, we Combine Like Terms on the left side: $150=10x$.
- Finally, we divide both sides of the equation by $10$ to get the solution $x=15$.
Oh, but it's his mom's $60\text{th}$ birthday. $15$ diamonds aren't enough, he thinks. Now he has to look for some other possibilities...
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Decide which equation best describes each of the given situations.
HintsFirst, determine the information we already know, and then assign the variable $x$ to the unknown value.
For example, if one rocket crocodile costs $5$ gold bars $4$ rocket crocodiles cost $4\times 5=20$ gold bars.
You just have to multiply the number of rocket crocodiles by the price per crocodile.
Don't forget Dr. Evil sells his island for $50$ gold bars.
SolutionTo solve word problems, you must recognize what values are given and what is unknown.
To write an equation, we assign the the variable $x$ to our unknown value.
From left to the right: Unknown number of diamonds
- The number of diamonds is unknown, but we know the price. So $x$ diamonds cost $10\times x$ gold bars. Write this term on the right side of the equation.
- By selling one island and 20 rocket crocodiles, Dr. Evil earns $50\times 1+5\times 20$ gold bars.
- This leads to the equation $50\times 1+5\times 20= 10\times x$.
- This time, we assign the variable $x$ to the number of rocket crocodiles. Together with the price of the island, we get $50\times 1+5\times x$.
- On the other side of the equation, we have the cost for $60$ diamonds at $10$ gold bars each: $10\times 60$.
- This leads to the equation $50\times 1+5\times x=10\times 60$.
- On the left side we have $50\times 1$ for the island plus $x\times 55$ for the rocket crocodiles, giving us $50\times 1+x\times 55$.
- On the other side of the equation, we have the cost for $60$ diamonds at $10$ gold bars each: $10\times 60$.
- This leads to the equation $50\times 1+x\times 55=10\times 60$.
-
Determine how many hours Dr. Evil's assistant can work.
HintsSolve the equation step-by-step:
- Simplify the equation by using PEMDAS.
- Isolate the variable using opposite operations.
The opposite operations are:
- $\text{addition}~\longleftrightarrow~\text{subtraction}$
- $\text{multiplication}~\longleftrightarrow~\text{division}$
SolutionUnfortunately, one of the rocket crocodiles exploded, and Dr. Evil's lab is completely destroyed.
Assistants are very expensive, $4$ gold coins an hour. The number of hours is unknown. So we assign the variable $x$ to this unknown number.
Once again Dr. Evil has to sell some things:
- his beach hut for $400$ gold coins
- $40$ piranhas for $3$ gold coins each
On the other side of the equation, Dr. Evil gets:
- $400\times 1$ gold coins for his beach hut
- $3\times 40$ gold coins for $40$ piranhas
- All together he has $400\times 1+3\times 40$ gold coins.
- $400\times 1+3\times 40=4\times x$
- By multiplying, we get: $400+120=4x$
- Next, we Combine Like Terms on the left side: $520=4x$.
- Last but not least, we divide both sides of the equation by $4$ to get $x=130$.
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Set up equations and solve for the unknown values.
HintsFirst, assign the variable $x$ to the unknown.
Formulate an equation with the things Lara sells on one side of the equal sign and the total money she needs for the teacups on the other side.
Use PEMDAS to simplify the equation.
To isolate the variable $x$, you have to perform opposite operations.
SolutionHow can we solve word problems?
- First, we assign the variable $x$ to the unknown.
- Next, we formulate an equation.
- Finally, we solve the equation using opposite operations.
$\begin{array}{rclcl} 29\times x &=& 1.50\times 30+2.50\times 40 & &|\text{ PEMDAS}\\ 29x&=&45+100\\ 29x&=&~145 & &|\text{ Combine Like Terms}\\ \color{#669900}{\div29}&&\color{#669900}{\div29}& &| (\times~\leftrightarrow~\div) \text{ Opposite Operations}\\ x&=&~~~~~5 \end{array}$
$x=5$ means Lara can buy $5$ new teacups.
Second Example In the second example, the price of each book is unknown. We have $1.50\times 30+x\times 40$ on one side and the cost of 6 new teacups on the other side: $37.50\times 6$. Now we have enough information to solve the equation.
$\begin{array}{rclcl} 37.50\times 6&=&2\times 30+4\times x&&|\text{PEMDAS}\\ 225&=&45+40x\\ \color{#669900}{-45}&&\color{#669900}{-45}&&| (+~\leftrightarrow~-) \text{ Opposite Operations}\\ 180&=&~40x\\ \color{#669900}{\div40}&&\color{#669900}{\div40}&&| (\times~\leftrightarrow~\div) \text{ Opposite Operations}\\ x&=&~4.5 \end{array}$
If Lara sells each book for $\$4.50$, she can buy the $6$ new teacups she wants.
Third Example In our last example, the number of books is unknown. We have $1.50\times 30+4\times x$ on one side and the cost of 6 new teacups on the other: $37.5\times 6$. Let's solve it!
$\begin{array}{rclcl} 25\times 8&=&2\times 30+4\times x&&|\text{PEMDAS}\\ 200&=&60+4x\\ \color{#669900}{-60}&&\color{#669900}{-60}&&| (+~\leftrightarrow~-) \text{ Opposite Operations}\\ 140&=&~~~4x\\ \color{#669900}{\div4}&=&\color{#669900}{\div4}&&| (\times~\leftrightarrow~\div) \text{ Opposite Operations}\\ x&=&~~~35 \end{array}$
$x=35$ means Lara has to sell $35$ books.
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Summarize what you know about opposite operations.
HintsTo solve an equation like $x+3=7$, you must isolate the variable:
$\begin{array}{rclcl} x+3&=&~~~7\\ \color{#669900}{-3}&&\color{#669900}{-3}&&| (+~\leftrightarrow~-) \text{ Opposite Operations}\\ x&=&~~~4. \end{array}$
Think of an equation like a scale in balance.
Removing weight on one side leads to an imbalance, so to stay balanced, you will need to remove it on the other side, too.
To solve an equation, the goal is to get a solution for the variable. That means that you should isolate the variable on one side of the equation.
SolutionWe can use Opposite Operations to isolate the variable $x$.
What are opposite operations?
- The opposite operation of addition is subtraction, and vice versa.
- The opposite operation of multiplication is division, and vice versa.
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Solve the following equations.
HintsIf you want to solve an equation, do it step-by-step:
- Simplify the equation using PEMDAS or the Distributive Property
- Isolate the Variable using Opposite Operations*
Opposite operations are
- $\text{addition}~\longleftrightarrow~\text{subtraction}$
- $\text{multiplication}~\longleftrightarrow~\text{division}$.
Some solutions do not equal $-1$, $1$ or $2$.
SolutionFor every equation, we first combine like terms using PEMDAS. Then we isolate the variable $x$ using opposite operations.
Equation One
$\begin{array}{rclcl} 5x-3x+2&=&~~~4\\ 2x+2&=&~~~4&&|\text{ Combine Like Terms}\\ \color{#669900}{-2}&&\color{#669900}{-2}&&| (+~\leftrightarrow~-) \text{ Opposite Operations}\\ 2x&=&~~~2\\ \color{#669900}{\div2}&&\color{#669900}{\div2}&&| (\times~\leftrightarrow~\div) \text{ Opposite Operations}\\ x&=&~~~2 \end{array}$
Equation Two
$\begin{array}{rclcl} 2x+3\times(x-1)&=&~~~7&&|\text{ Distributive Property}\\ 2x+3x-3&&~~~7\\ 5x-3&=&~~~7&&|\text{ Combine Like Terms}\\ \color{#669900}{+3}&&\color{#669900}{+3}&&| (-~\leftrightarrow~+) \text{ Opposite Operations}\\ 5x&=&~10\\ \color{#669900}{\div5}&&\color{#669900}{\div5}&&| (\times~\leftrightarrow~\div) \text{ Opposite Operations}\\ x&=&~~~2 \end{array}$
Equation Three
$\begin{array}{rclcl} 5x+2&=&~~~4\times(x+1)\\ 5x+2&&4x+4&&|\text{ Distributive Property}\\ \color{#669900}{-2}&&\color{#669900}{-2}&&| (+~\leftrightarrow~-) \text{ Opposite Operations}\\ 5x&=&4x+2\\ \color{#669900}{-4x}&&\color{#669900}{-4x}&&| (+~\leftrightarrow~-) \text{ Opposite Operations}\\ x&=&~~~2 \end{array}$
Equation Four
$\begin{array}{rclcl} 2x+4&=&4\times(x-1)+10&&|\text{ Distributive Property}\\ 2x+4&=&4x-4+10\\ 2x+4&=&4x+6&&|\text{ Combine Like Terms}\\ \color{#669900}{-4}&&\color{#669900}{-4}&&| (+~\leftrightarrow~-) \text{ Opposite Operations}\\ 2x&=&4x+2\\ \color{#669900}{-4x}&&\color{#669900}{-4x}&&| (+~\leftrightarrow~-) \text{ Opposite Operations}\\ -2x&=&~~~2\\ \color{#669900}{\div(-2)}&&\color{#669900}{\div(-2)}&&| (\times~\leftrightarrow~\div) \text{ Opposite Operations}\\ x&=&-1 \end{array}$
The other two equations have solutions that do not equal $-1$, $1$ or $2$.
