11 answers
The equation \(2x + 3y = 39\) is a linear equation. It can be rearranged into the slope-intercept form \(y = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept.
To determine if it is a function, we can express \(y\) in terms of \(x\):
\[ 3y = 39 - 2x \] \[ y = \frac{39 - 2x}{3} \]
This shows that for each value of \(x\), there is exactly one corresponding value of \(y\). Therefore, it is a function.
So, the answer is: Linear; function.
The equation \(y = 3x^2 + 2\) is not a linear equation because the term \(3x^2\) includes a variable raised to a power greater than one (specifically, \(x\) is squared).
To analyze whether it is a function, we observe that for each value of \(x\), there is exactly one corresponding value of \(y\). Therefore, it meets the definition of a function.
So, the answer is: Nonlinear; function.
To determine whether the given data represents a linear or nonlinear relationship, or if it defines a function, we can analyze the points provided:
The points are:
To check if the relationship is linear, we can look at the differences in \(y\) values for equal differences in \(x\) values.
Calculating the differences:
From \((-2, -7)\) to \((-1, -4)\): \[ y: -4 - (-7) = 3 \quad (x: -1 - (-2) = 1) \]
From \((-1, -4)\) to \((0, -1)\): \[ y: -1 - (-4) = 3 \quad (x: 0 - (-1) = 1) \]
From \((0, -1)\) to \((1, 2)\): \[ y: 2 - (-1) = 3 \quad (x: 1 - 0 = 1) \]
From \((1, 2)\) to \((2, 5)\): \[ y: 5 - 2 = 3 \quad (x: 2 - 1 = 1) \]
The change in \(y\) is consistent (3) for each increment of \(x\) (1) across all points. This constancy suggests that the relationship is linear.
Thus, because each \(x\) value corresponds to exactly one \(y\) value, the relationship is a function.
Therefore, the answer is: Linear; function.
To determine whether the relationship represented by the given points is linear or nonlinear, and whether it defines a function, let's first examine the points provided:
To analyze the relationship, we can observe the \(y\) values and their changes relative to the \(x\) values:
From (0, 1) to (1, 3): \[ y: 3 - 1 = 2 \quad (x: 1 - 0 = 1) \]
From (1, 3) to (2, 9): \[ y: 9 - 3 = 6 \quad (x: 2 - 1 = 1) \]
From (2, 9) to (3, 27): \[ y: 27 - 9 = 18 \quad (x: 3 - 2 = 1) \]
From (3, 27) to (4, 81): \[ y: 81 - 27 = 54 \quad (x: 4 - 3 = 1) \]
The differences in \(y\) are \(2\), \(6\), \(18\), and \(54\), which are not constant. Instead, the \(y\) values appear to follow a pattern of increasing rapidly, suggesting a nonlinear relationship.
Each \(x\) value corresponds to exactly one \(y\) value. Therefore, this relationship does meet the definition of a function.
Since the relationship is nonlinear and it is a function, the answer is: Nonlinear; function.
To determine whether the given set of ordered pairs represents a function, we need to check if each input (i.e., the first element in each pair, corresponding to \(x\)) has a unique output (the second element, corresponding to \(y\)). A relation is considered a function if no two ordered pairs have the same first value.
The given set of pairs is:
Now, let's check the \(x\) values in each pair:
Next, we look for duplicate \(x\) values:
Since each \(x\) value is associated with exactly one \(y\) value, the relationship does indeed show a function.
So, the answer is: Yes, it is a function.
It looks like you have provided \(x\) values, the corresponding calculations for \(y = 2x + 3\), the \(y\) values that result from the calculations, and the resulting ordered pairs \((x, y)\).
Let's verify the computations and summaries for clarity:
For \(x = -5\): \[ y = 2(-5) + 3 = -10 + 3 = -7 \quad \text{(Pair: } (-5, -7)\text{)} \]
For \(x = -3\): \[ y = 2(-3) + 3 = -6 + 3 = -3 \quad \text{(Pair: } (-3, -3)\text{)} \]
For \(x = 0\): \[ y = 2(0) + 3 = 0 + 3 = 3 \quad \text{(Pair: } (0, 3)\text{)} \]
For \(x = 1\): \[ y = 2(1) + 3 = 2 + 3 = 5 \quad \text{(Pair: } (1, 5)\text{)} \]
For \(x = 2\): \[ y = 2(2) + 3 = 4 + 3 = 7 \quad \text{(Pair: } (2, 7)\text{)} \]
For \(x = 4\): \[ y = 2(4) + 3 = 8 + 3 = 11 \quad \text{(Pair: } (4, 11)\text{)} \]
The ordered pairs generated from the function \(y = 2x + 3\) are:
Is it a function? Yes, every \(x\) value corresponds to exactly one \(y\) value, fulfilling the definition of a function.
Is it linear? Yes, the equation \(y = 2x + 3\) is in the form of a linear equation, which can be represented as a straight line graph.
