4. Equations
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Chapter 5
4.
Equations
This lesson delves into the complexities of equations, emphasizing the importance of variables, solutions to an equation, and mathematical modeling. It aims to simplify these mathematical tools by connecting them to real-life situations. For instance, equations can be used to calculate costs in group activities or to determine physical characteristics like height and weight. By making these connections, the guide helps both students and professionals understand how to apply equations in practical, everyday contexts.
| Student Learning Objectives: |
|---|
|
| | 10 Theory slides |
| | 8 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Equations
Slide of 10
Problems or situations in daily life can often be represented by algebraic expressions. When an expression is set equal to another expression or number, it is called an equation. This lesson introduce equations and how to use them to model real-life situations.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
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Challenge
Finding the Correct Floor
Tadeo and his little brother Dylan are in an elevator. Dylan randomly presses buttons on the panel.
External credits: Icongeek26
The elevator goes 3 floors up and then 7 floors down before arriving at the ground floor, or Floor 1. Which floor were Tadeo and his brother on when they got on the elevator?
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Discussion
Equation
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Discussion
Solution of an Equation
A solution of an equation is a value that makes the equation true. It satisfies the equation when it is substituted for the variable of the equation. Consider the following equation. x+1=4 The solution of the equation is x=3 because 3 is the only value of x that makes the equation true.
For the other values that are not solutions, the right-hand side of the solution is not equal to the left-hand side. This is shown by anot equalsign ≠. Consider the result of substituting x=1 into the equation. Some equations are have no solution. For example, x=x+1 can never be true because a number cannot equal more than one of itself. x=x+1 * Equations can have multiple or even infinitely many solutions. x+x=2x This equation will be true for all values of x since the expressions on the left- and right-hand sides are equivalent expressions.
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Example
Deciding What to Order
A restaurant has four different combos meals.
a
Suppose two customers order the same combo. The price of their combo satisfies the equation 2x+8=22. Which combo did they choose?
{"type":"choice","form":{"alts":["Combo I","Combo II","Combo III","Combo IV"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":0}
b
Now suppose they order different combos. The prices of the combos they choose satisfy the equation x^2-15x=-54. Select the combos they chose.
{"type":"multichoice","form":{"alts":["Combo I","Combo II","Combo III","Combo IV"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":[1,3]}
Hint
a
Substitute the given prices into the equation. Which combo makes the equation true?
b
Substitute the given prices into the equation. Which combos make the equation true?
Solution
a
To find which combo the two customers chose, we substitute each combo price into the equation and check whether it satisfies the equation. Let's start with the price of Combo I, which is $ 7.
Since x=7 makes the equation true, it is a solution to the equation. Next, we substitute the remaining combo prices and simplify to check whether the equation has any other solutions.
| Menu | Menu Price ($) | Substitution into 2x+8=22 | Check |
|---|---|---|---|
| Combo I | 7 | 2( 7)+8? =22 | 22 = 22 ✓ |
| Combo II | 9 | 2( 9)+8? =22 | 26 ≠ 22 * |
| Combo III | 5 | 2( 5)+8? =22 | 18 ≠ 22 * |
| Combo IV | 6 | 2( 6)+8? =22 | 20 ≠ 22 * |
Only the price of Combo I satisfies the equation, so they ordered Combo I.
b
To find which combos they chose, we use the same method as in Part A. We substitute each combo price into the equation and check whether it satisfies the equation. Let's start with the price of Combo I.
x^2-54x=15
Substitute
x= 7
( 7)^2-15( 7) ? =-54
CalcPow
Calculate power
49-15(7)? =-54
Multiply
Multiply
49-105? =-54
SubTerms
Subtract terms
-56 ≠ -54
Since x=7 does not satisfy the given equation, it is not a solution. Neither customer ordered Combo I. Next, we check the remaining combo prices.
| Menu | Menu Price | Substitution into x^2-15x=-54 | Check |
|---|---|---|---|
| Combo I | 7 | ( 7)^2-15( 7)? =-54 49-105? =-54 | -56 ≠ -54 * |
| Combo II | 9 | ( 9)^2-15( 9)? =-54 81-135 ? =-54 | -54 = -54 ✓ |
| Combo III | 5 | ( 5)^2-15( 5)? =-54 25-75 ? =-54 | -50 ≠ -54 * |
| Combo IV | 6 | ( 6)^2-15( 6)? =-54 36-90 ? =-54 | -54 = -54 ✓ |
We found that x = 9 and x = 6 are solutions to the equation. Since they ordered different combos, one customer ordered Combo II, and the other ordered Combo IV.
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Pop Quiz
Checking Solutions
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Example
Writing Equations
Write an equation for each situation.
a
A bookshelf is 12 centimeters taller than a cabinet. The bookshelf is 168 centimeters tall. Write an equation to find the height of the cabinet.
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b
A smaller box weighs three-fourths as much as the larger box. The larger box weighs 52 kilograms. Write an equation to find the weight of the smaller box.
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c
The difference between the ages of an oak tree and a maple tree is 3 years. The oak tree is older and is 16 years old. Write an equation to find the age of the maple tree.
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Hint
a
The bookshelf is 12 centimeters taller than the cabinet. This involves adding some numbers.
b
Three-fourths of a number is the same as the product of 34 and that number.
c
Difference
indicates subtraction.
Solution
a
To write the equation for the situation, let's start by writing it in words. Then, we can look at smaller phrases and write them as algebraic expressions.
The word is represents an equals sign, which means that the expressions to the left and right of this word are equal. We also know that the bookshelf's height is 168 centimeters.
Next, more than stands for addition. We can also use a variable x to represent the height of the cabinet.
Saying that the height of the cabinet is 12 centimeters less than the bookshelf's height is another way of saying the same thing. Here, less than stands for subtraction. x=168-12 Both equations can be used to find the height of the cabinet.
b
Three-fourths of a number means the product of 34 and that number. With this in mind, let's write the sentence as an equation using x for the weight of the smaller box.
We can also write the fraction 34 as 75100 or 0.75. This means that, for example, the equation x=0.75 * 52 also represents the given sentence.
c
Here, the word difference
stands for subtraction. Let's use x for the age of the maple tree and translate the words into an equation.
Another way to state the situation is to say that the oak tree is 3 years older than the maple tree. Then, its age is the maple tree's age plus 3. x+3=16 Both equations can be used to find the age of the maple tree. A situation can be represented by more than one equation. We can rewrite an equation in different ways by rearranging its terms.
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Example
Buying Games
Dylan and Tadeo visit a game store.
a
Dylan purchases 7 new premium trading cards. After this purchase, he has 25 premium cards. How many premium cards did Dylan have before he bought the new ones?
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b
After buying 3 used games for $ 15, Tadeo is left with $ 32 in his wallet. How much money did he have before he bought the games?
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Hint
a
The number of cards Dylan had before going into the store is 7 less than 25.
b
Tadeo started with $15 more than the $32 he has left after buying the games.
Solution
a
Dylan bought 7 new cards and now has 25 in total. This means he had 7 fewer cards before buying more. Let n represent the number of cards he had before buying more, and write an equation to find n.
n= 25- 7 Now we simplify the equation to find how many cards he had.
The solution to the equation is n=18. This means that Dylan had 18 premium cards before buying more.
b
The amount of money Tadeo started with is the total of the $ 15 he spent on the games and the $ 32 he had left after the purchase. Let's use m to represent the amount of money he started with and write an equation for this situation.
m= 32+ 15 Now we can simplify the equation.
We found that Tadeo had $ 47 before buying the games.
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Example
Selling Old Toys
Mark and Kevin are selling their old toy cars at a secondhand store.
a
The store pays them $ 36 for 12 cars. If each car is worth the same amount, how much do they get for one car?
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b
How many cars do the boys need to sell to earn $54?
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Hint
a
To find the amount paid for one car, divide $36 by 12.
b
Use the amount from Part A. Write a division equation to represent the situation.
Solution
a
The store paid the boys the same amount for each toy car. This means we can find x, the amount they got for one car, by dividing the total of $ 36 they received by the 12 cars they sold.
x = 36/12 ⇒ x= 3 The solution to the equation is x=3, so the shop paid the boys $ 3 for each toy car sold.
b
In Part A, we found that the store pays $ 3 for each car. To find the number of cars, y, they need to sell to get $ 54, we divide 54 by 3.
y = 54/3 ⇒ y = 18 The boys need to sell 18 toy cars.
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Closure
Finding the Correct Floor
At the beginning of the lesson, Dylan randomly pressed some elevator buttons.
External credits: Icongeek26
The elevator went 3 floors up and then 7 floors down before arriving at the ground floor, or Floor 1. Which floor were Tadeo and his brother on when they got on the elevator?
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Hint
Try to
undotheir trip.
Solution
Let's think about this problem backwards. We can use a variable x for the floor where the boys got into the elevator.
After going 3 floors up and then 7 floors down, they ended up on Floor 1. If we reverse their trip, they go 7 floors up from Floor 1 and then 3 floors down, arriving at Floor x. We can use this information to write an equation. x = 1 + 7 - 3 Now we can simplify this equation to find the floor the boys were on when they got on the elevator.
The solution to the equation is x=5. This means that the boys got on the elevator on Floor 5.
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Equations
Exercises
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{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["y"],"constants":[]},"rendered":false},"text":""},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["y+6=13","y=7","13=y+6"]}}
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When writing words as equations, we look for keywords such as is
or equals
to find the placement of the equals sign.
5 less than a number b equals 9.
↓
=
Another keyword in our sentence is less than,
which indicates subtraction.
5 less than a number b equals 9.
⇓ ↓
- =
Now let's rewrite the other terms 5, the number b, and 9 to complete our equation.
The equation b-5=9 represents the given sentence.
Let's start by examining the given sentence again. Note that the word sum
means addition.
The sum of a number y and 3 is 13.
⇓
+
Next, we place the equals sign where the keyword is
stands.
The sum of a number y and 6 is 13.
⇓ ⇓
+ =
Finally write the terms y, 6, and 13 to complete the equation.
The equation y+6=13 represents the given word sentence.
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Solution
Consider the following equation. x-7=31 Choose the sentence that does not represent this equation.
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We will decide which option does not represent the given equation. Let's start by examining the given options.
| Sentences |
|---|
| 7 subtracted from a number is 31. |
| A number decreased by 7 is 31. |
| A number minus 7 is the same as 31. |
| A number is 7 less than 31. |
| 31 equals the difference between a number and 7. |
Let's write each sentence as an equation and check whether it matches the given equation. When writing sentences as equations, we look for keywords such as is
or equals
to find the placement of the equals sign. There are also some other keywords to decide the place of the operational signs. In this case, we want to use subtraction, so let's look for keywords that indicate subtraction.
difference
less than
subtracted from
decreased by
Since all of the options use the variable x, we will also use x to indicate a number
. With this in mind, let's write one of the options as an equation.
We can write the other sentences as equations by following the same steps.
| Sentences | Equation |
|---|---|
| 7 subtracted from a number is 31. | x - 7 = 31 |
| A number decreased by 7 is 31. | x - 7 = 31 |
| A number minus 7 is the same as 31. | x - 7 = 31 |
| A number is 7 less than 31. | x = 31 - 7 |
| 31 equals the difference between a number and 7. | 31 = x - 7 |
As we can see from the table, the sentence A number is 7 less than 31.
needs to be represented with a different equation.
x=31-7
is not the same as
x-7=31
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Solution
A group of 6 friends are planning to have a fun day together. They will choose one out of three different activities. The equations in the table represent the number of people x who can participate in each activity for $18.
| Activity | Cost ($ ) |
|---|---|
| Balloon Ride | 3x-42=18 |
| Amusement Park | 5x-22=18 |
| Zoo Trip | 2x+6=18 |
Which activity should the friends choose if they want to spend exactly $18 and all do the same activity?
{"type":"choice","form":{"alts":["Balloon Ride","Amusement Park","Zoo Trip"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":2}
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We are asked to determine which activity the friends should choose if they want to spend exactly $18 and all do the same activity. There are 6 people in the friends group, so let's substitute x=6 into each equation and see which of them produce true statements. Let's start with the equation for the balloon ride.
Notice that x=6 is not a solution to this equation, so the 6 friends cannot join a balloon ride for $18. Let's apply the same thought process and substitute x=6 into the equations for the amusement park and zoo trip.
| Activity | Cost ($) | Substitute x=6 | Multiply | Add or Subtract | Is x=6 a Solution? |
|---|---|---|---|---|---|
| Balloon Ride | 3x-42=18 | 3( 6)-42? =18 | 18-42? =18 | -24=18 * | No |
| Amusement Park | 5x-22=18 | 5( 6)-22? =18 | 30-22? =18 | 8 ≠ 18 * | No |
| Zoo Trip | 2x+6=18 | 2( 6)+6? =18 | 12+6? =18 | 18=18 ✓ | Yes |
We found that x=6 is a solution to the equation that represents the cost of the zoo trip. This means that if the friends want to spend exactly $18 and all do something together, they should choose the zoo trip.
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Solution
A scale balances with four violet blocks on one side and one orange x-block on the other side. This situation is represented by the following equation. 4=x Heichi places 16 more violet blocks to the left side. The scale is now unbalanced.
In an attempt to balance the scale again, he placed more orange x-blocks on the right side and wrote the following equation that represents the balanced scale. 20=? * x Find the number that makes the equation true.
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We know that the following scale is balanced when four violet blocks are on the left side and one orange x-block on the right side.
We want to find a number that balances the scale again when multiplied by x.
Let's keep adding 4 violet blocks to the left side and 1 orange x-block to the other side until we get 20 violet blocks. This method will keep the scale balanced!
As we can see, if there are 20 violet blocks on the left side of the scale, we need 5 orange x-blocks to keep the scale balanced. This means that we need to multiply the x-blocks by 5. Now we can complete the given equation. 20=5* x
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