Solved on Jan 18, 2024

Determine if each equation is Linear, Exponential, or Neither: A) $y=-5x-2$ (Linear), B) $y=4x^2+5$ (Neither), C) $y=-2+3/4x$ (Neither), D) $y=(1.5)^x$ (Exponential), E) $y=2(5)^2$ (Exponential), F) $y=5(x-4)^2$ (Neither), G) $y=2-5(0.6)^x$ (Exponential), H) $y=\sqrt{4x+3}$ (Neither), M) $4=7-5x$ (Linear), N) $y=4x+5x^2$ (Neither), O) $y=5(2.4)^x$ (Exponential), P) $y=2x-4^x$ (Neither).

STEP 1

Assumptions
1. A linear equation is of the form $y = mx + b$ , where $m$ and $b$ are constants.
2. An exponential equation is of the form $y = a \cdot b^x$ , where $a$ and $b$ are constants, and $b$ is the base of the exponential function.
3. A quadratic equation is of the form $y = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants.
4. Equations that do not fit the form of linear or exponential are considered neither.

STEP 2

Identify the type of equation A: $y=-5x-2$ .
This equation is in the form $y = mx + b$ with $m = -5$ and $b = -2$ , which is a linear equation.

STEP 3

Identify the type of equation B: $y=4x^2+5$ .
This equation is in the form $y = ax^2 + bx + c$ with $a = 4$ , $b = 0$ , and $c = 5$ , which is a quadratic equation, so it is neither linear nor exponential.

STEP 4

Identify the type of equation C: $y=-2+\frac{3}{4}x$ .
This equation is in the form $y = mx + b$ with $m = \frac{3}{4}$ and $b = -2$ , which is a linear equation.

STEP 5

Identify the type of equation D: $y=(1.5)^x$ .
This equation is in the form $y = a \cdot b^x$ with $a = 1$ and $b = 1.5$ , which is an exponential equation.

STEP 6

Identify the type of equation E: $y=2(5)^2$ .
This equation simplifies to $y = 2 \cdot 25 = 50$ , which is a constant function. Since it does not have the variable $x$ , it is neither linear nor exponential.

STEP 7

Identify the type of equation F: $y=5(x-4)^2$ .
This equation is in the form $y = a(x - h)^2 + k$ with $a = 5$ , $h = 4$ , and $k = 0$ , which is a quadratic equation, so it is neither linear nor exponential.

STEP 8

Identify the type of equation G: $y=2-5(0.6)^x$ .
This equation is in the form $y = a - b \cdot c^x$ with $a = 2$ , $b = 5$ , and $c = 0.6$ , which is an exponential equation.

STEP 9

Identify the type of equation H: $y=\sqrt{4x+3}$ .
This equation is not in the form of a linear or exponential equation and involves a square root function, so it is neither.

STEP 10

Identify the type of equation M: $4=7-5x$ .
This equation can be rewritten in the form $y = mx + b$ by isolating $y$ : $y = -5x + 7$ , which is a linear equation.

STEP 11

Identify the type of equation N: $y=4x+5x^2$ .
This equation is in the form $y = ax^2 + bx + c$ with $a = 5$ , $b = 4$ , and $c = 0$ , which is a quadratic equation, so it is neither linear nor exponential.

STEP 12

Identify the type of equation O: $y=5(2.4)^x$ .
This equation is in the form $y = a \cdot b^x$ with $a = 5$ and $b = 2.4$ , which is an exponential equation.

STEP 13

Identify the type of equation P: $y=2x-4^x$ .
This equation is not in the form of a linear or exponential equation and involves a term with an exponential function as well as a linear term, so it is neither.
To summarize: - A: Linear - B: Neither (Quadratic) - C: Linear - D: Exponential - E: Neither (Constant) - F: Neither (Quadratic) - G: Exponential - H: Neither (Square Root) - M: Linear - N: Neither (Quadratic) - O: Exponential - P: Neither

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STEP 1

STEP 2

Identify the type of equation A: $y=-5x-2$ .
This equation is in the form $y = mx + b$ with $m = -5$ and $b = -2$ , which is a linear equation.

STEP 3

Identify the type of equation B: $y=4x^2+5$ .
This equation is in the form $y = ax^2 + bx + c$ with $a = 4$ , $b = 0$ , and $c = 5$ , which is a quadratic equation, so it is neither linear nor exponential.

STEP 4

Identify the type of equation C: $y=-2+\frac{3}{4}x$ .
This equation is in the form $y = mx + b$ with $m = \frac{3}{4}$ and $b = -2$ , which is a linear equation.

STEP 5

Identify the type of equation D: $y=(1.5)^x$ .
This equation is in the form $y = a \cdot b^x$ with $a = 1$ and $b = 1.5$ , which is an exponential equation.

STEP 6

Identify the type of equation E: $y=2(5)^2$ .
This equation simplifies to $y = 2 \cdot 25 = 50$ , which is a constant function. Since it does not have the variable $x$ , it is neither linear nor exponential.

STEP 7

Identify the type of equation F: $y=5(x-4)^2$ .
This equation is in the form $y = a(x - h)^2 + k$ with $a = 5$ , $h = 4$ , and $k = 0$ , which is a quadratic equation, so it is neither linear nor exponential.

STEP 8

Identify the type of equation G: $y=2-5(0.6)^x$ .
This equation is in the form $y = a - b \cdot c^x$ with $a = 2$ , $b = 5$ , and $c = 0.6$ , which is an exponential equation.

STEP 9

Identify the type of equation H: $y=\sqrt{4x+3}$ .
This equation is not in the form of a linear or exponential equation and involves a square root function, so it is neither.

STEP 10

Identify the type of equation M: $4=7-5x$ .
This equation can be rewritten in the form $y = mx + b$ by isolating $y$ : $y = -5x + 7$ , which is a linear equation.

STEP 11

Identify the type of equation N: $y=4x+5x^2$ .
This equation is in the form $y = ax^2 + bx + c$ with $a = 5$ , $b = 4$ , and $c = 0$ , which is a quadratic equation, so it is neither linear nor exponential.

STEP 12

Identify the type of equation O: $y=5(2.4)^x$ .
This equation is in the form $y = a \cdot b^x$ with $a = 5$ and $b = 2.4$ , which is an exponential equation.

STEP 13

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STEP 1

STEP 2

Identify the type of equation A: y=−5x−2y=-5x-2y=−5x−2.This equation is in the form y=mx+by = mx + by=mx+b with m=−5m = -5m=−5 and b=−2b = -2b=−2, which is a linear equation.

STEP 3

Identify the type of equation B: y=4x2+5y=4x^2+5y=4x2+5.This equation is in the form y=ax2+bx+cy = ax^2 + bx + cy=ax2+bx+c with a=4a = 4a=4, b=0b = 0b=0, and c=5c = 5c=5, which is a quadratic equation, so it is neither linear nor exponential.

STEP 4

Identify the type of equation C: y=−2+34xy=-2+\frac{3}{4}xy=−2+43​x.This equation is in the form y=mx+by = mx + by=mx+b with m=34m = \frac{3}{4}m=43​ and b=−2b = -2b=−2, which is a linear equation.

STEP 5

Identify the type of equation D: y=(1.5)xy=(1.5)^xy=(1.5)x.This equation is in the form y=a⋅bxy = a \cdot b^xy=a⋅bx with a=1a = 1a=1 and b=1.5b = 1.5b=1.5, which is an exponential equation.

STEP 6

Identify the type of equation E: y=2(5)2y=2(5)^2y=2(5)2.This equation simplifies to y=2⋅25=50y = 2 \cdot 25 = 50y=2⋅25=50, which is a constant function. Since it does not have the variable xxx, it is neither linear nor exponential.

STEP 7

Identify the type of equation F: y=5(x−4)2y=5(x-4)^2y=5(x−4)2.This equation is in the form y=a(x−h)2+ky = a(x - h)^2 + ky=a(x−h)2+k with a=5a = 5a=5, h=4h = 4h=4, and k=0k = 0k=0, which is a quadratic equation, so it is neither linear nor exponential.

STEP 8

Identify the type of equation G: y=2−5(0.6)xy=2-5(0.6)^xy=2−5(0.6)x.This equation is in the form y=a−b⋅cxy = a - b \cdot c^xy=a−b⋅cx with a=2a = 2a=2, b=5b = 5b=5, and c=0.6c = 0.6c=0.6, which is an exponential equation.

STEP 9

Identify the type of equation H: y=4x+3y=\sqrt{4x+3}y=4x+3​.This equation is not in the form of a linear or exponential equation and involves a square root function, so it is neither.

STEP 10

Identify the type of equation M: 4=7−5x4=7-5x4=7−5x.This equation can be rewritten in the form y=mx+by = mx + by=mx+b by isolating yyy: y=−5x+7y = -5x + 7y=−5x+7, which is a linear equation.

STEP 11

Identify the type of equation N: y=4x+5x2y=4x+5x^2y=4x+5x2.This equation is in the form y=ax2+bx+cy = ax^2 + bx + cy=ax2+bx+c with a=5a = 5a=5, b=4b = 4b=4, and c=0c = 0c=0, which is a quadratic equation, so it is neither linear nor exponential.

STEP 12

Identify the type of equation O: y=5(2.4)xy=5(2.4)^xy=5(2.4)x.This equation is in the form y=a⋅bxy = a \cdot b^xy=a⋅bx with a=5a = 5a=5 and b=2.4b = 2.4b=2.4, which is an exponential equation.

STEP 13

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Identify the type of equation A: $y=-5x-2$ .
This equation is in the form $y = mx + b$ with $m = -5$ and $b = -2$ , which is a linear equation.

Identify the type of equation B: $y=4x^2+5$ .
This equation is in the form $y = ax^2 + bx + c$ with $a = 4$ , $b = 0$ , and $c = 5$ , which is a quadratic equation, so it is neither linear nor exponential.

Identify the type of equation C: $y=-2+\frac{3}{4}x$ .
This equation is in the form $y = mx + b$ with $m = \frac{3}{4}$ and $b = -2$ , which is a linear equation.

Identify the type of equation D: $y=(1.5)^x$ .
This equation is in the form $y = a \cdot b^x$ with $a = 1$ and $b = 1.5$ , which is an exponential equation.

Identify the type of equation E: $y=2(5)^2$ .
This equation simplifies to $y = 2 \cdot 25 = 50$ , which is a constant function. Since it does not have the variable $x$ , it is neither linear nor exponential.

Identify the type of equation F: $y=5(x-4)^2$ .
This equation is in the form $y = a(x - h)^2 + k$ with $a = 5$ , $h = 4$ , and $k = 0$ , which is a quadratic equation, so it is neither linear nor exponential.

Identify the type of equation G: $y=2-5(0.6)^x$ .
This equation is in the form $y = a - b \cdot c^x$ with $a = 2$ , $b = 5$ , and $c = 0.6$ , which is an exponential equation.

Identify the type of equation H: $y=\sqrt{4x+3}$ .
This equation is not in the form of a linear or exponential equation and involves a square root function, so it is neither.

Identify the type of equation M: $4=7-5x$ .
This equation can be rewritten in the form $y = mx + b$ by isolating $y$ : $y = -5x + 7$ , which is a linear equation.

Identify the type of equation N: $y=4x+5x^2$ .
This equation is in the form $y = ax^2 + bx + c$ with $a = 5$ , $b = 4$ , and $c = 0$ , which is a quadratic equation, so it is neither linear nor exponential.

Identify the type of equation O: $y=5(2.4)^x$ .
This equation is in the form $y = a \cdot b^x$ with $a = 5$ and $b = 2.4$ , which is an exponential equation.