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## Lesson 4: Equations and Their Solutions
### 4.1: What is a Solution?
Let's explore the given problem in a structured and detailed manner.
### Problem Statement
A granola bite contains 27 calories. Most of the calories come from [tex]\( c \)[/tex] grams of carbohydrates. The rest come from other ingredients. One gram of carbohydrate contains 4 calories.
The equation [tex]\( 4c + 5 = 27 \)[/tex] represents this relationship. Let's break down and solve this equation.
### Question 1: What could the 5 represent in this situation?
In the equation [tex]\( 4c + 5 = 27 \)[/tex]:
- The term [tex]\( 4c \)[/tex] represents the calories from [tex]\( c \)[/tex] grams of carbohydrates since 1 gram of carbohydrate contains 4 calories.
- The number 5 represents the calories from other ingredients in the granola bite that are not carbohydrates.
Thus, the 5 in the equation represents the calories contributed by these other ingredients.
### Question 2: Priya said that neither 8 nor 3 could be the solution to the equation. Explain why she is correct.
To verify why 8 and 3 are not solutions, we will substitute these values into the equation and see if they satisfy it.
1. Checking [tex]\( c = 8 \)[/tex]:
Substitute [tex]\( c = 8 \)[/tex] into the equation [tex]\( 4c + 5 = 27 \)[/tex]:
[tex]\[ 4(8) + 5 = 32 + 5 = 37 \neq 27 \][/tex]
Since 37 is not equal to 27, 8 is not a solution.
2. Checking [tex]\( c = 3 \)[/tex]:
Substitute [tex]\( c = 3 \)[/tex] into the equation [tex]\( 4c + 5 = 27 \)[/tex]:
[tex]\[ 4(3) + 5 = 12 + 5 = 17 \neq 27 \][/tex]
Since 17 is not equal to 27, 3 is not a solution.
Therefore, Priya is correct that neither 8 nor 3 could be the solution to the equation.
### Question 3: Find the solution to the equation.
To solve the equation [tex]\( 4c + 5 = 27 \)[/tex], we follow these steps:
1. Subtract 5 from both sides of the equation:
[tex]\[ 4c + 5 - 5 = 27 - 5 \][/tex]
Simplifying both sides, we get:
[tex]\[ 4c = 22 \][/tex]
2. Divide both sides by 4 to isolate [tex]\( c \)[/tex]:
[tex]\[ c = \frac{22}{4} \][/tex]
Simplifying the division, we get:
[tex]\[ c = 5.5 \][/tex]
So, the solution to the equation [tex]\( 4c + 5 = 27 \)[/tex] is [tex]\( c = 5.5 \)[/tex]. This means that the granola bite contains 5.5 grams of carbohydrates.
### 4.1: What is a Solution?
Let's explore the given problem in a structured and detailed manner.
### Problem Statement
A granola bite contains 27 calories. Most of the calories come from [tex]\( c \)[/tex] grams of carbohydrates. The rest come from other ingredients. One gram of carbohydrate contains 4 calories.
The equation [tex]\( 4c + 5 = 27 \)[/tex] represents this relationship. Let's break down and solve this equation.
### Question 1: What could the 5 represent in this situation?
In the equation [tex]\( 4c + 5 = 27 \)[/tex]:
- The term [tex]\( 4c \)[/tex] represents the calories from [tex]\( c \)[/tex] grams of carbohydrates since 1 gram of carbohydrate contains 4 calories.
- The number 5 represents the calories from other ingredients in the granola bite that are not carbohydrates.
Thus, the 5 in the equation represents the calories contributed by these other ingredients.
### Question 2: Priya said that neither 8 nor 3 could be the solution to the equation. Explain why she is correct.
To verify why 8 and 3 are not solutions, we will substitute these values into the equation and see if they satisfy it.
1. Checking [tex]\( c = 8 \)[/tex]:
Substitute [tex]\( c = 8 \)[/tex] into the equation [tex]\( 4c + 5 = 27 \)[/tex]:
[tex]\[ 4(8) + 5 = 32 + 5 = 37 \neq 27 \][/tex]
Since 37 is not equal to 27, 8 is not a solution.
2. Checking [tex]\( c = 3 \)[/tex]:
Substitute [tex]\( c = 3 \)[/tex] into the equation [tex]\( 4c + 5 = 27 \)[/tex]:
[tex]\[ 4(3) + 5 = 12 + 5 = 17 \neq 27 \][/tex]
Since 17 is not equal to 27, 3 is not a solution.
Therefore, Priya is correct that neither 8 nor 3 could be the solution to the equation.
### Question 3: Find the solution to the equation.
To solve the equation [tex]\( 4c + 5 = 27 \)[/tex], we follow these steps:
1. Subtract 5 from both sides of the equation:
[tex]\[ 4c + 5 - 5 = 27 - 5 \][/tex]
Simplifying both sides, we get:
[tex]\[ 4c = 22 \][/tex]
2. Divide both sides by 4 to isolate [tex]\( c \)[/tex]:
[tex]\[ c = \frac{22}{4} \][/tex]
Simplifying the division, we get:
[tex]\[ c = 5.5 \][/tex]
So, the solution to the equation [tex]\( 4c + 5 = 27 \)[/tex] is [tex]\( c = 5.5 \)[/tex]. This means that the granola bite contains 5.5 grams of carbohydrates.
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