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: |
|---|
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| | 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?
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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 the sentence as an algebraic expression.
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 to describe the same situation. 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. For example, the equation x=0.75 * 52 also represents the given sentence.
c
Here, the word difference
means subtraction. Let's use x for the age of the maple tree and translate the words into an equation.
Another way to describe the situation is to say that the oak tree is 3 years older than the maple tree or that its age is the age of the maple tree 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.
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.
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
Exercise 3.1
When he is halfway to the beach, half of the strawberries are gone. He walks 4 more blocks, then stops for a moment to check his phone before he walks the last block to the beach. He reaches his destination 20 minutes after he began walking. At this point, three-eighths of his strawberries are left.
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{"type":"choice","form":{"alts":["Yes","No"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":1}
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Let's analyze the information we are given to see if we can determine how many blocks there are between the store and the beach. We are told that after walking halfway, Heichi walks another 4 blocks. He then checks his phone and walks the last 1 block to the beach. Let's make a drawing that represents the given information about blocks.
Notice that the second half of the walk to the beach consists of 4+1=5 blocks. This means that there are the same number of blocks for the first half of the walk, too. 4+1+4+1=10 In total, Heichi was 10 blocks away from the beach when he bought the basket of strawberries. It took him 20 minutes to walk these 10 blocks. We can use this information to find the time it takes him to walk each block. Since it takes the same amount of time to walk each block, let's divide 20 by 10 to find the time it takes to walk each block. 20/10=2 This means that Heichi walks each block in 2 minutes. We can say that yes, there is enough information to find the time it takes Heichi to walk each block.
This time, we will try to find the number of strawberries Heichi ate during the time he walked last block. We know that he started with 32 strawberries and that he had eaten half of them when he was halfway through the walk. Half the strawberries=32/2=16 He eats 16 strawberries during the first half of the walk. When he reaches the beach, three-eighths of the strawberries are left. Let's calculate three-eights of the strawberries by multiplying 32 and 38.
This means that when he arrives at the beach, he only has 12 strawberries left. Let's make a diagram that represents the information about the strawberries.
According to diagram, the numbers of strawberries eaten are different in the first and second halves of the walk. First Half: & 32-16=16 First Half: & 16-12=4 There is no information given about whether Heichi eats the same number of strawberries over each block for the second half of the walk. We can estimate the number of berries eaten in each block, but there is not enough information to find the exact number of strawberries that he eats while walking the last block.
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