Question
Plug in (3,10) and (0,1) and check to show that the system of equations has
more than one solution. Explain your work.
more than one solution. Explain your work.
Answers
GPT-4o mini
To determine whether the system of equations defined by the points (3, 10) and (0, 1) has more than one solution, we first need to establish a relationship between these points.
1. **Identify the equation of the line**: We can derive the linear equation that passes through the points (3, 10) and (0, 1).
- **Find the slope (m)**:
The slope \( m \) is calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{10 - 1}{3 - 0} = \frac{9}{3} = 3.
\]
- **Use point-slope form**:
Now that we have the slope, we can use one of the points (let's use (0, 1)) to find the line equation in point-slope form, which is:
\[
y - y_1 = m(x - x_1).
\]
Plugging in (0, 1):
\[
y - 1 = 3(x - 0) \implies y - 1 = 3x \implies y = 3x + 1.
\]
2. **Verify the other point**: Now we check if the other point (3, 10) satisfies this equation:
\[
y = 3(3) + 1 = 9 + 1 = 10.
\]
Since (3, 10) lies on the line defined by \( y = 3x + 1 \), this confirms that both points define the same linear equation.
3. **Does the system have more than one solution?**: The equation \( y = 3x + 1 \) represents an infinite number of solutions because it is a linear equation. Any value of \( x \) will produce a corresponding value of \( y \) on the line. Therefore, there are indeed more than one solution to the system, as any point satisfying this linear equation is a solution.
4. **Conclusion**: Thus, the system of equations, defined by the points (3, 10) and (0, 1), has an infinite number of solutions represented by the line \( y = 3x + 1 \).
1. **Identify the equation of the line**: We can derive the linear equation that passes through the points (3, 10) and (0, 1).
- **Find the slope (m)**:
The slope \( m \) is calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{10 - 1}{3 - 0} = \frac{9}{3} = 3.
\]
- **Use point-slope form**:
Now that we have the slope, we can use one of the points (let's use (0, 1)) to find the line equation in point-slope form, which is:
\[
y - y_1 = m(x - x_1).
\]
Plugging in (0, 1):
\[
y - 1 = 3(x - 0) \implies y - 1 = 3x \implies y = 3x + 1.
\]
2. **Verify the other point**: Now we check if the other point (3, 10) satisfies this equation:
\[
y = 3(3) + 1 = 9 + 1 = 10.
\]
Since (3, 10) lies on the line defined by \( y = 3x + 1 \), this confirms that both points define the same linear equation.
3. **Does the system have more than one solution?**: The equation \( y = 3x + 1 \) represents an infinite number of solutions because it is a linear equation. Any value of \( x \) will produce a corresponding value of \( y \) on the line. Therefore, there are indeed more than one solution to the system, as any point satisfying this linear equation is a solution.
4. **Conclusion**: Thus, the system of equations, defined by the points (3, 10) and (0, 1), has an infinite number of solutions represented by the line \( y = 3x + 1 \).