Question
A. Select the two equations that represent this situation if x represents the age of the younger brother and y represents the age of the older brother.
(2 points)
Responses
x+y=6
𝑥
+
𝑦
=
6
x plus y is equal to 6
y=2+3x2
𝑦
=
2
+
3
𝑥
2
y is equal to 2 plus 3 x squared
y=6+x
𝑦
=
6
+
𝑥
y is equal to 6 plus x
y=2+3x
𝑦
=
2
+
3
𝑥
y is equal to 2 plus 3 x
Question 2
B. Solve the system from part A by graphing. Clearly label the solution(s) on the graph and then state the solution in words.
(2 points)
Responses
x+y=6
𝑥
+
𝑦
=
6
x plus y is equal to 6
y=2+3x2
𝑦
=
2
+
3
𝑥
2
y is equal to 2 plus 3 x squared
y=6+x
𝑦
=
6
+
𝑥
y is equal to 6 plus x
y=2+3x
𝑦
=
2
+
3
𝑥
y is equal to 2 plus 3 x
Question 2
B. Solve the system from part A by graphing. Clearly label the solution(s) on the graph and then state the solution in words.
Answers
Answered by
GPT-4o mini
To identify which equations represent the situation where \(x\) is the age of the younger brother and \(y\) is the age of the older brother, let's analyze the options given:
1. **\(x + y = 6\)**: This indicates that the sum of the younger brother's age and the older brother's age is 6, which makes sense in this context.
2. **\(y = 2 + 3x^2\)**: This is a quadratic equation. Since it doesn't represent a linear relationship between ages, it's less likely to fit the situation.
3. **\(y = 6 + x\)**: This suggests that the older brother is always 6 years older than the younger brother, which could also make sense.
4. **\(y = 2 + 3x\)**: This suggests that the older brother's age is determined by a linear function of the younger brother's age, which could be reasonable if there is a specific age gap.
From the analysis, the two equations that represent the situation are:
1. \(x + y = 6\) (Equation 1)
2. \(y = 6 + x\) (Equation 3) - but here we notice it suggests a fixed gap that might be less intuitively reasonable.
Therefore, the better choice for relationships based on common scenarios where one brother's age is dependent on the other's is:
1. **\(x + y = 6\)**
2. **\(y = 2 + 3x\)**
### B. Solve the system by graphing:
Let's graph the equations \(x + y = 6\) and \(y = 2 + 3x\).
1. **For \(x + y = 6\)**:
- When \(x = 0\), \(y = 6\): The point is (0, 6).
- When \(y = 0\), \(x = 6\): The point is (6, 0).
- This line goes from (0, 6) to (6, 0).
2. **For \(y = 2 + 3x\)**:
- When \(x = 0\), \(y = 2\): The point is (0, 2).
- When \(x = 1\), \(y = 5\): The point is (1, 5).
- This line goes through the points (0, 2) and (1, 5).
#### Graphing the Equations
To sketch the graph:
- **Plot the points** from the calculations above.
- Draw the line for each equation.
#### Finding the Intersection
The solution to the system is where the two lines intersect. In this case, solving for \(y\) we can substitute \(y\) from the second equation into the first:
1. Substitute: \(x + (2 + 3x) = 6\)
- Combine like terms: \(4x + 2 = 6\)
- Subtract 2: \(4x = 4\)
- Divide by 4: \(x = 1\)
2. Substitute \(x = 1\) back into either equation to find \(y\):
- From \(y = 2 + 3(1) = 5\), we find \(y = 5\).
Thus, the solution is:
**\(x = 1\) (younger brother's age) and \(y = 5\) (older brother's age)**.
### Solution Summary
The younger brother is 1 year old and the older brother is 5 years old.
1. **\(x + y = 6\)**: This indicates that the sum of the younger brother's age and the older brother's age is 6, which makes sense in this context.
2. **\(y = 2 + 3x^2\)**: This is a quadratic equation. Since it doesn't represent a linear relationship between ages, it's less likely to fit the situation.
3. **\(y = 6 + x\)**: This suggests that the older brother is always 6 years older than the younger brother, which could also make sense.
4. **\(y = 2 + 3x\)**: This suggests that the older brother's age is determined by a linear function of the younger brother's age, which could be reasonable if there is a specific age gap.
From the analysis, the two equations that represent the situation are:
1. \(x + y = 6\) (Equation 1)
2. \(y = 6 + x\) (Equation 3) - but here we notice it suggests a fixed gap that might be less intuitively reasonable.
Therefore, the better choice for relationships based on common scenarios where one brother's age is dependent on the other's is:
1. **\(x + y = 6\)**
2. **\(y = 2 + 3x\)**
### B. Solve the system by graphing:
Let's graph the equations \(x + y = 6\) and \(y = 2 + 3x\).
1. **For \(x + y = 6\)**:
- When \(x = 0\), \(y = 6\): The point is (0, 6).
- When \(y = 0\), \(x = 6\): The point is (6, 0).
- This line goes from (0, 6) to (6, 0).
2. **For \(y = 2 + 3x\)**:
- When \(x = 0\), \(y = 2\): The point is (0, 2).
- When \(x = 1\), \(y = 5\): The point is (1, 5).
- This line goes through the points (0, 2) and (1, 5).
#### Graphing the Equations
To sketch the graph:
- **Plot the points** from the calculations above.
- Draw the line for each equation.
#### Finding the Intersection
The solution to the system is where the two lines intersect. In this case, solving for \(y\) we can substitute \(y\) from the second equation into the first:
1. Substitute: \(x + (2 + 3x) = 6\)
- Combine like terms: \(4x + 2 = 6\)
- Subtract 2: \(4x = 4\)
- Divide by 4: \(x = 1\)
2. Substitute \(x = 1\) back into either equation to find \(y\):
- From \(y = 2 + 3(1) = 5\), we find \(y = 5\).
Thus, the solution is:
**\(x = 1\) (younger brother's age) and \(y = 5\) (older brother's age)**.
### Solution Summary
The younger brother is 1 year old and the older brother is 5 years old.