Question
Write 10 things that you know to be true. Try to solve and graph the solution.
Function of the Day: 2(x + 4) = 16
(You can think of the x as a crate of rocks or a candy bar and the constants as rocks or dollars)
Answers
Answered by
GPT-4o mini
Sure! Let's start with the function of the day, which is the equation:
\[ 2(x + 4) = 16. \]
### Step 1: Solve the Equation
1. **Distribute the 2:**
\[
2x + 8 = 16.
\]
2. **Subtract 8 from both sides:**
\[
2x = 16 - 8,
\]
\[
2x = 8.
\]
3. **Divide both sides by 2:**
\[
x = \frac{8}{2},
\]
\[
x = 4.
\]
So, \( x = 4 \) is the solution.
### Step 2: Graph the Solution
To graph the equation \( 2(x + 4) = 16 \), we can rearrange it into a more workable form, like \( y = 2(x + 4) \) and plot it along with \( y = 16 \).
1. Rearranging \( y = 2(x + 4) \):
\[
y = 2x + 8.
\]
2. The line \( y = 16 \) is a horizontal line at \( y = 16 \).
3. Graph both lines:
- **Line 1 (y = 2x + 8)**: This is a straight line with a slope of 2 and y-intercept at 8.
- **Line 2 (y = 16)**: A horizontal line which intersects the y-axis at 16.
### Graph:
To visualize:
- **Line 1:** Starts at (0, 8) and goes up 2 units for every 1 unit it goes to the right. You can plot additional points like (1, 10), (2, 12), and so on.
- **Line 2:** This is a constant line across the height of 16 on the y-axis.
The point where these two lines intersect is the solution \( x = 4 \).
**Coordinate of Intersection:** If \( x = 4 \):
\[
y = 2(4) + 8 = 8 + 8 = 16.
\]
Thus, the lines intersect at the point (4, 16).
### Summary of the Solution
- The solution to the equation \( 2(x + 4) = 16 \) is \( x = 4 \).
- The graphical representation includes a line for the equation \( y = 2x + 8 \) and a horizontal line at \( y = 16 \), intersecting at (4, 16).
### 10 Statements That Are True
1. The sun rises in the east and sets in the west.
2. Water freezes at 0 degrees Celsius.
3. Earth revolves around the Sun, taking approximately 365.25 days for one complete orbit.
4. All living things are composed of cells.
5. The speed of light in a vacuum is approximately 299,792 kilometers per second.
6. Humans require oxygen to live.
7. The Earth is the third planet from the Sun.
8. Gravity is a force that attracts two bodies towards each other.
9. A triangle has three sides.
10. The chemical symbol for water is H₂O.
\[ 2(x + 4) = 16. \]
### Step 1: Solve the Equation
1. **Distribute the 2:**
\[
2x + 8 = 16.
\]
2. **Subtract 8 from both sides:**
\[
2x = 16 - 8,
\]
\[
2x = 8.
\]
3. **Divide both sides by 2:**
\[
x = \frac{8}{2},
\]
\[
x = 4.
\]
So, \( x = 4 \) is the solution.
### Step 2: Graph the Solution
To graph the equation \( 2(x + 4) = 16 \), we can rearrange it into a more workable form, like \( y = 2(x + 4) \) and plot it along with \( y = 16 \).
1. Rearranging \( y = 2(x + 4) \):
\[
y = 2x + 8.
\]
2. The line \( y = 16 \) is a horizontal line at \( y = 16 \).
3. Graph both lines:
- **Line 1 (y = 2x + 8)**: This is a straight line with a slope of 2 and y-intercept at 8.
- **Line 2 (y = 16)**: A horizontal line which intersects the y-axis at 16.
### Graph:
To visualize:
- **Line 1:** Starts at (0, 8) and goes up 2 units for every 1 unit it goes to the right. You can plot additional points like (1, 10), (2, 12), and so on.
- **Line 2:** This is a constant line across the height of 16 on the y-axis.
The point where these two lines intersect is the solution \( x = 4 \).
**Coordinate of Intersection:** If \( x = 4 \):
\[
y = 2(4) + 8 = 8 + 8 = 16.
\]
Thus, the lines intersect at the point (4, 16).
### Summary of the Solution
- The solution to the equation \( 2(x + 4) = 16 \) is \( x = 4 \).
- The graphical representation includes a line for the equation \( y = 2x + 8 \) and a horizontal line at \( y = 16 \), intersecting at (4, 16).
### 10 Statements That Are True
1. The sun rises in the east and sets in the west.
2. Water freezes at 0 degrees Celsius.
3. Earth revolves around the Sun, taking approximately 365.25 days for one complete orbit.
4. All living things are composed of cells.
5. The speed of light in a vacuum is approximately 299,792 kilometers per second.
6. Humans require oxygen to live.
7. The Earth is the third planet from the Sun.
8. Gravity is a force that attracts two bodies towards each other.
9. A triangle has three sides.
10. The chemical symbol for water is H₂O.
Answer
give me 10 things u know to be true about this equation
Answered by
GPT-4o mini
Sure! Here are 10 statements that can be made about the equation \( 2(x + 4) = 16 \):
1. **Linear Equation**: The equation is a linear equation, which means it represents a straight line when graphed.
2. **Single Variable**: The equation contains only one variable, \( x \).
3. **Coefficient**: The coefficient of \( x \) is \( 2 \), indicating that for every unit increase in \( x \), the left-hand side increases by \( 2 \).
4. **Constant Term**: The constant term in the equation is \( 8 \), derived from distributing the \( 2 \) across \( (x + 4) \).
5. **Solution Exists**: The equation has a solution; specifically, it has exactly one solution due to it being a linear equation.
6. **Value of \( x \)**: The solution for \( x \) is \( 4 \), as found by isolating \( x \) in the original equation.
7. **Graphical Intersection**: The solution \( x = 4 \) corresponds to an intersection point on the graph of the lines represented by \( y = 2(x + 4) \) and \( y = 16 \).
8. **Equation Balance**: If you add or subtract the same value from both sides of the equation, the equality remains true, maintaining balance.
9. **Basic Algebraic Manipulation**: The equation can be simplified using basic algebraic operations such as distribution, subtraction, and division.
10. **Real Number Solution**: The solution to the equation is a real number (\( 4 \)), which means it lies on the real number line and can be interpreted in various practical contexts.
These statements provide insights into the mathematical nature of the equation as well as its properties and solution.
1. **Linear Equation**: The equation is a linear equation, which means it represents a straight line when graphed.
2. **Single Variable**: The equation contains only one variable, \( x \).
3. **Coefficient**: The coefficient of \( x \) is \( 2 \), indicating that for every unit increase in \( x \), the left-hand side increases by \( 2 \).
4. **Constant Term**: The constant term in the equation is \( 8 \), derived from distributing the \( 2 \) across \( (x + 4) \).
5. **Solution Exists**: The equation has a solution; specifically, it has exactly one solution due to it being a linear equation.
6. **Value of \( x \)**: The solution for \( x \) is \( 4 \), as found by isolating \( x \) in the original equation.
7. **Graphical Intersection**: The solution \( x = 4 \) corresponds to an intersection point on the graph of the lines represented by \( y = 2(x + 4) \) and \( y = 16 \).
8. **Equation Balance**: If you add or subtract the same value from both sides of the equation, the equality remains true, maintaining balance.
9. **Basic Algebraic Manipulation**: The equation can be simplified using basic algebraic operations such as distribution, subtraction, and division.
10. **Real Number Solution**: The solution to the equation is a real number (\( 4 \)), which means it lies on the real number line and can be interpreted in various practical contexts.
These statements provide insights into the mathematical nature of the equation as well as its properties and solution.