Let's address each question one by one:
Question 1:
The linear equation is given as y = 0.2x + 15.
- True: The linear equation is correctly represented as y = 0.2x + 15.
- False: The average increase in temperature each year is 0.2°C, not 1°C (this is the slope of the line).
- True: In the year 2010 (which is x=10), the temperature would be: \[ y = 0.2(10) + 15 = 2 + 15 = 17°C. \]
- True: If the trend continues, since the slope is 0.2°C per year, the temperature in 2030 (which is x=30) will be: \[ y = 0.2(30) + 15 = 6 + 15 = 21°C. \] The temperature in 2029 (x=29) will be: \[ y = 0.2(29) + 15 = 5.8 + 15 = 20.8°C. \] Thus, the temperature in 2030 will be \( 21°C \), which is \( 0.2°C \) higher than in 2029 but not 1°C higher. So this statement is False.
- False: The average increase in temperature every 5 years is: \[ 0.2°C/year \times 5 years = 1°C. \] Therefore, this statement is True.
True statements from Question 1:
- y = 0.2x + 15 represents the linear equation.
- In the year 2010 the temperature was 17°C.
- The average increase in temperature every 5 years is 1°C.
Question 2:
We need to find equations that have the solution set of {14, -5}.
- True: \((4x - 1)(x + 5) = 0\) gives \(x = \frac{1}{4}\) and \(x = -5\).
- False: \((x - 14)(x - 5) = 0\) gives \(x = 14\) and \(x = 5\).
- False: \((2x - 8)(x - 5) = 0\) gives \(x = 4\) and \(x = 5\).
- False: \((x + 14)(x + 5) = 0\) gives \(x = -14\) and \(x = -5\).
- False: \((8x - 2)(x + 5) = 0\) gives \(x = \frac{1}{4}\) and \(x = -5\).
- False: \((4x + 2)(x + 5) = 0\) gives \(x = -\frac{1}{2}\) and \(x = -5\).
- True: \((x - 14)(x + 5) = 0\) gives \(x = 14\) and \(x = -5\).
Equations with the solution set of {14, -5}:
- \((4x - 1)(x + 5) = 0\)
- \((x - 14)(x + 5) = 0\)
Question 3:
The equation is \(f(x) = x^2 - 5x + 6\).
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The solutions to the equation \(f(x) = x^2 - 5x + 6\) when \(f(x) = 6\) are the points where the graph intersects the line \(y=6\). This occurs at the points corresponding to the roots of the equation, which can be calculated by setting \(f(x) = 6\): \[ x^2 - 5x + 6 - 6 = 0 \Rightarrow x^2 - 5x = 0 \Rightarrow x(x - 5) = 0 \Rightarrow x = 0, 5. \]
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The solutions to the equation \(f(x) = x^2 - 5x + 6\) when \(f(2)\) and \(f(3)\) correspond to the output of the function at these inputs:
- \(f(2) = 2^2 - 5(2) + 6 = 4 - 10 + 6 = 0.\)
- \(f(3) = 3^2 - 5(3) + 6 = 9 - 15 + 6 = 0.\)
Thus, the points when \(f(x) = 0\) occur at \( x=2, 3\).
Points for completion about \( f(x) = x^2 - 5x + 6 \):
- When \(f(x) = 6\) → the points are (0, 6) and (5, 6).
- When \(f(2)\) and \(f(3)\) → the points are (2, 0) and (3, 0).
I hope this clarifies the questions and provides accurate responses!
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