To address your questions systematically, let’s start from the given tables.
Question 1: Finding the Equations
Table 1:
x | y
-------
-3 | 4
6 | -2
9 | -4
To find the equation of the line that fits these points, we can calculate the slope (m) and y-intercept (b) using the two-point form. Let's use the points (-3, 4) and (6, -2).
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Finding the slope (m): \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - 4}{6 - (-3)} = \frac{-6}{9} = -\frac{2}{3} \]
-
Finding the y-intercept (b): We'll use the point (-3, 4) to find b: \[ y = mx + b \] \[ 4 = -\frac{2}{3}(-3) + b \] \[ 4 = 2 + b \] \[ b = 2 \]
Hence, the equation for the first table is: \[ y = -\frac{2}{3}x + 2 \]
Table 2:
x | y
-------
7 | 6
5 | 3
1 | -3
Using the points (7, 6) and (5, 3):
-
Finding the slope (m): \[ m = \frac{3 - 6}{5 - 7} = \frac{-3}{-2} = \frac{3}{2} \]
-
Finding the y-intercept (b): Using the point (7, 6): \[ 6 = \frac{3}{2}(7) + b \] \[ 6 = \frac{21}{2} + b \] \[ b = 6 - \frac{21}{2} = 6 - 10.5 = -4.5 \]
Thus, the equation for the second table is: \[ y = \frac{3}{2}x - 4.5 \]
Question 2: Solving the System of Equations
The system of equations derived from Question 1 is:
- \( y = -\frac{2}{3}x + 2 \)
- \( y = \frac{3}{2}x - 4.5 \)
We set them equal to each other to solve for x:
\[-\frac{2}{3}x + 2 = \frac{3}{2}x - 4.5\]
Multiplying through by 6 to eliminate the fractions: \[-4x + 12 = 9x - 27\]
Bringing like terms together: \[12 + 27 = 9x + 4x\] \[39 = 13x\] \[x = 3\]
Substituting x back into one of the equations to find y: Using \(y = -\frac{2}{3}(3) + 2\): \[y = -2 + 2 = 0\]
Thus, the solution to the system of equations is: \[(3, 0)\]
Question 3: Real-life Problem Representation
Imagine you are analyzing the relationship between the hours studied and test scores for two students, Alice and Bob.
- Alice's study hours can be modeled by the equation \(y = -\frac{2}{3}x + 2\), where x represents the hours spent studying and y represents the predicted test score.
- Bob's study hours can be modeled by the equation \(y = \frac{3}{2}x - 4.5\).
Finding the intersection (the point (3, 0)) would represent a situation where both students studied for 3 hours, leading to a predicted score of 0 (indicating they didn't meet the threshold for passing).
Question 4: System of Inequalities
To write a system of inequalities based on some given graph, you would typically identify constraints represented by the lines that form boundaries. Suppose your graph has constraints along the y-axis and x-axis in the first quadrant below the line \(y = mx + b\).
- \(y \leq mx + b\) (Area below the line)
- \(y \geq 0\) (Area above the x-axis)
- \(x \geq 0\) (Area to the right of the y-axis)
Please note that without a specific graph provided, the inequalities may vary based on visual representation. The specific inequalities depend on the slopes and intercepts of the lines drawn.
Let me know if you need more specific information!