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Determine whether the equation represents a linear or nonlinear function. Explain in a complete sentence.

y = x + 5
a) Linear; because it has a square root term.
b) Nonlinear; because it has a square root term.
c) Linear; because it has a constant added.
d) Nonlinear; because it has a constant added.
y = 4x - 2
a) Linear; because it is in the form y = mx + b.
b) Nonlinear; because it is in the form y = mx + b.
c) Linear; because it has a coefficient of 4.
d) Nonlinear; because it has a coefficient of 4.
y = 9x
a) Linear; because it has a coefficient of 9.
b) Nonlinear; because it has a coefficient of 9.
c) Linear; because it is in the form y = mx.
d) Nonlinear; because it is in the form y = mx.
y = (x - 1)(x+7)
a) Linear; because it is a product of two linear functions.
b) Nonlinear; because it is a product of two linear functions.
c) Linear; because it is a quadratic function.
d) Nonlinear; because it is a quadratic function.

1 Answer

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Final answer:

y = x + 5, y = 4x - 2, and y = 9x are linear equations as they fit the form y = mx + b, where m and b represent the slope and y-intercept, respectively. y = (x - 1)(x+7) is nonlinear because it is a quadratic equation when expanded.

Step-by-step explanation:

To determine whether the given equations represent a linear or nonlinear function, we need to check if they are in the standard form of a linear equation, which is y = a + bx. In this form, b is the slope, and a is the y-intercept. A linear equation graphs as a straight line.

  • y = x + 5 is a linear equation because it is in the form y = mx + b, with m = 1 (the coefficient of x) and b = 5 (the constant term). So, the correct answer is c) Linear; because it has a constant added.
  • y = 4x - 2 is a linear equation because it is in the form y = mx + b, with m = 4 (the coefficient of x) and b = -2 (the constant term). So, the correct choice is a) Linear; because it is in the form y = mx + b.
  • y = 9x is a linear equation because it can also be written in the form y = mx + b, with m = 9 and b = 0. Thus, the correct answer is c) Linear; because it is in the form y = mx.
  • y = (x - 1)(x+7) is a nonlinear function because when you expand this, it becomes a quadratic equation, which has the form y = ax² + bx + c, where a ≠ 0. Therefore, the correct choice is d) Nonlinear; because it is a quadratic function.
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