Solved on Jan 18, 2024

Classify equations as Linear, Exponential, or Neither: $y=-5x-2$ (Linear), $y=4x^2+5$ (Quadratic), $y=-2+3/4x$ (Rational), $y=(1.5)^x$ (Exponential), $g\approx2(5)^2$ (Exponential), $y=5(x-4)^2$ (Quadratic), $y=2-5(0.6)^x$ (Exponential), $y=\sqrt{4x+3}$ (Square Root), $y=17-5x$ (Linear), $y=7x+18x^3$ (Polynomial), $y=(1.23)^x$ (Exponential), $y=(x+4)^2+3$ (Quadratic), $y=7-5x$ (Linear), $y=4x+5x^2$ (Polynomial), $y=5(2.4)^x$ (Exponential), $y=2x-4^x$ (Exponential).

STEP 1

Assumptions
1. Linear equations are of the form $y = mx + b$ , where $m$ and $b$ are constants.
2. Exponential equations are of the form $y = a \cdot b^x$ , where $a$ and $b$ are constants, and $b$ is not equal to 1.
3. Equations that do not fit the form of either linear or exponential are classified as neither.

STEP 2

Analyze equation A: $y = -5x - 2$
This equation is in the form $y = mx + b$ where $m = -5$ and $b = -2$ , which fits the definition of a linear equation.
Solution for A: Linear

STEP 3

Analyze equation B: $y = 4x^2 + 5$
This equation is a quadratic equation, not linear or exponential, because it contains an $x^2$ term.
Solution for B: Neither

STEP 4

Analyze equation C: $y = -2 + \frac{3}{4}x$
This equation can be rewritten as $y = \frac{3}{4}x - 2$ , which is in the form $y = mx + b$ where $m = \frac{3}{4}$ and $b = -2$ , fitting the definition of a linear equation.
Solution for C: Linear

STEP 5

Analyze equation D: $y = (1.5)^x$
This equation is in the form $y = a \cdot b^x$ where $a = 1$ and $b = 1.5$ , which fits the definition of an exponential equation.
Solution for D: Exponential

STEP 6

Analyze equation E: $g \approx 2(5)^2$
This equation is a constant because $5^2$ is a constant and multiplying by 2 does not change that. It is neither linear nor exponential.
Solution for E: Neither

STEP 7

Analyze equation F: $y = 5(x - 4)^2$
This equation is a quadratic equation, not linear or exponential, because it contains a squared term $(x - 4)^2$ .
Solution for F: Neither

STEP 8

Analyze equation G: $y = 2 - 5(0.6)^x$
This equation is in the form $y = a - b^x$ where $a = 2$ and $b = 0.6$ , which fits the definition of an exponential equation.
Solution for G: Exponential

STEP 9

Analyze equation H: $y = \sqrt{4x + 3}$
This equation involves a square root and is neither linear nor exponential.
Solution for H: Neither

STEP 10

Analyze equation I: $y = 17 - 5x$
This equation is in the form $y = mx + b$ where $m = -5$ and $b = 17$ , fitting the definition of a linear equation.
Solution for I: Linear

STEP 11

Analyze equation J: $y = 7x + 18x^3$
This equation contains a cubic term $18x^3$ and is neither linear nor exponential.
Solution for J: Neither

STEP 12

Analyze equation K: $y = (1.23)^x$
This equation is in the form $y = b^x$ where $b = 1.23$ , fitting the definition of an exponential equation.
Solution for K: Exponential

STEP 13

Analyze equation L: $y = (x + 4)^2 + 3$
This equation is a quadratic equation, not linear or exponential, because it contains a squared term $(x + 4)^2$ .
Solution for L: Neither

STEP 14

Analyze equation M: $y = 7 - 5x$
This equation is in the form $y = mx + b$ where $m = -5$ and $b = 7$ , fitting the definition of a linear equation.
Solution for M: Linear

STEP 15

Analyze equation N: $y = 4x + 5x^2$
This equation contains a quadratic term $5x^2$ and is neither linear nor exponential.
Solution for N: Neither

STEP 16

Analyze equation O: $y = 5(2.4)^x$
This equation is in the form $y = a \cdot b^x$ where $a = 5$ and $b = 2.4$ , fitting the definition of an exponential equation.
Solution for O: Exponential

STEP 17

Analyze equation P: $y = 2x - 4^x$
This equation contains both a linear term $2x$ and an exponential term $-4^x$ , so it does not fit neatly into either category.
Solution for P: Neither

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STEP 1

STEP 2

Analyze equation A: $y = -5x - 2$
This equation is in the form $y = mx + b$ where $m = -5$ and $b = -2$ , which fits the definition of a linear equation.
Solution for A: Linear

STEP 3

Analyze equation B: $y = 4x^2 + 5$
This equation is a quadratic equation, not linear or exponential, because it contains an $x^2$ term.
Solution for B: Neither

STEP 4

Analyze equation C: $y = -2 + \frac{3}{4}x$
This equation can be rewritten as $y = \frac{3}{4}x - 2$ , which is in the form $y = mx + b$ where $m = \frac{3}{4}$ and $b = -2$ , fitting the definition of a linear equation.
Solution for C: Linear

STEP 5

Analyze equation D: $y = (1.5)^x$
This equation is in the form $y = a \cdot b^x$ where $a = 1$ and $b = 1.5$ , which fits the definition of an exponential equation.
Solution for D: Exponential

STEP 6

Analyze equation E: $g \approx 2(5)^2$
This equation is a constant because $5^2$ is a constant and multiplying by 2 does not change that. It is neither linear nor exponential.
Solution for E: Neither

STEP 7

Analyze equation F: $y = 5(x - 4)^2$
This equation is a quadratic equation, not linear or exponential, because it contains a squared term $(x - 4)^2$ .
Solution for F: Neither

STEP 8

Analyze equation G: $y = 2 - 5(0.6)^x$
This equation is in the form $y = a - b^x$ where $a = 2$ and $b = 0.6$ , which fits the definition of an exponential equation.
Solution for G: Exponential

STEP 9

Analyze equation H: $y = \sqrt{4x + 3}$
This equation involves a square root and is neither linear nor exponential.
Solution for H: Neither

STEP 10

Analyze equation I: $y = 17 - 5x$
This equation is in the form $y = mx + b$ where $m = -5$ and $b = 17$ , fitting the definition of a linear equation.
Solution for I: Linear

STEP 11

Analyze equation J: $y = 7x + 18x^3$
This equation contains a cubic term $18x^3$ and is neither linear nor exponential.
Solution for J: Neither

STEP 12

Analyze equation K: $y = (1.23)^x$
This equation is in the form $y = b^x$ where $b = 1.23$ , fitting the definition of an exponential equation.
Solution for K: Exponential

STEP 13

Analyze equation L: $y = (x + 4)^2 + 3$
This equation is a quadratic equation, not linear or exponential, because it contains a squared term $(x + 4)^2$ .
Solution for L: Neither

STEP 14

Analyze equation M: $y = 7 - 5x$
This equation is in the form $y = mx + b$ where $m = -5$ and $b = 7$ , fitting the definition of a linear equation.
Solution for M: Linear

STEP 15

Analyze equation N: $y = 4x + 5x^2$
This equation contains a quadratic term $5x^2$ and is neither linear nor exponential.
Solution for N: Neither

STEP 16

Analyze equation O: $y = 5(2.4)^x$
This equation is in the form $y = a \cdot b^x$ where $a = 5$ and $b = 2.4$ , fitting the definition of an exponential equation.
Solution for O: Exponential

STEP 17

Analyze equation P: $y = 2x - 4^x$
This equation contains both a linear term $2x$ and an exponential term $-4^x$ , so it does not fit neatly into either category.
Solution for P: Neither

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STEP 1

STEP 2

Analyze equation A: y=−5x−2y = -5x - 2y=−5x−2This equation is in the form y=mx+by = mx + by=mx+b where m=−5m = -5m=−5 and b=−2b = -2b=−2, which fits the definition of a linear equation.Solution for A: Linear

STEP 3

Analyze equation B: y=4x2+5y = 4x^2 + 5y=4x2+5This equation is a quadratic equation, not linear or exponential, because it contains an x2x^2x2 term.Solution for B: Neither

STEP 4

STEP 5

Analyze equation D: y=(1.5)xy = (1.5)^xy=(1.5)xThis equation is in the form y=a⋅bxy = a \cdot b^xy=a⋅bx where a=1a = 1a=1 and b=1.5b = 1.5b=1.5, which fits the definition of an exponential equation.Solution for D: Exponential

STEP 6

Analyze equation E: g≈2(5)2g \approx 2(5)^2g≈2(5)2This equation is a constant because 525^252 is a constant and multiplying by 2 does not change that. It is neither linear nor exponential.Solution for E: Neither

STEP 7

Analyze equation F: y=5(x−4)2y = 5(x - 4)^2y=5(x−4)2This equation is a quadratic equation, not linear or exponential, because it contains a squared term (x−4)2(x - 4)^2(x−4)2.Solution for F: Neither

STEP 8

Analyze equation G: y=2−5(0.6)xy = 2 - 5(0.6)^xy=2−5(0.6)xThis equation is in the form y=a−bxy = a - b^xy=a−bx where a=2a = 2a=2 and b=0.6b = 0.6b=0.6, which fits the definition of an exponential equation.Solution for G: Exponential

STEP 9

Analyze equation H: y=4x+3y = \sqrt{4x + 3}y=4x+3​This equation involves a square root and is neither linear nor exponential.Solution for H: Neither

STEP 10

Analyze equation I: y=17−5xy = 17 - 5xy=17−5xThis equation is in the form y=mx+by = mx + by=mx+b where m=−5m = -5m=−5 and b=17b = 17b=17, fitting the definition of a linear equation.Solution for I: Linear

STEP 11

Analyze equation J: y=7x+18x3y = 7x + 18x^3y=7x+18x3This equation contains a cubic term 18x318x^318x3 and is neither linear nor exponential.Solution for J: Neither

STEP 12

Analyze equation K: y=(1.23)xy = (1.23)^xy=(1.23)xThis equation is in the form y=bxy = b^xy=bx where b=1.23b = 1.23b=1.23, fitting the definition of an exponential equation.Solution for K: Exponential

STEP 13

Analyze equation L: y=(x+4)2+3y = (x + 4)^2 + 3y=(x+4)2+3This equation is a quadratic equation, not linear or exponential, because it contains a squared term (x+4)2(x + 4)^2(x+4)2.Solution for L: Neither

STEP 14

Analyze equation M: y=7−5xy = 7 - 5xy=7−5xThis equation is in the form y=mx+by = mx + by=mx+b where m=−5m = -5m=−5 and b=7b = 7b=7, fitting the definition of a linear equation.Solution for M: Linear

STEP 15

Analyze equation N: y=4x+5x2y = 4x + 5x^2y=4x+5x2This equation contains a quadratic term 5x25x^25x2 and is neither linear nor exponential.Solution for N: Neither

STEP 16

Analyze equation O: y=5(2.4)xy = 5(2.4)^xy=5(2.4)xThis equation is in the form y=a⋅bxy = a \cdot b^xy=a⋅bx where a=5a = 5a=5 and b=2.4b = 2.4b=2.4, fitting the definition of an exponential equation.Solution for O: Exponential

STEP 17

Analyze equation P: y=2x−4xy = 2x - 4^xy=2x−4xThis equation contains both a linear term 2x2x2x and an exponential term −4x-4^x−4x, so it does not fit neatly into either category.Solution for P: Neither

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Analyze equation A: $y = -5x - 2$
This equation is in the form $y = mx + b$ where $m = -5$ and $b = -2$ , which fits the definition of a linear equation.
Solution for A: Linear

Analyze equation B: $y = 4x^2 + 5$
This equation is a quadratic equation, not linear or exponential, because it contains an $x^2$ term.
Solution for B: Neither

Analyze equation D: $y = (1.5)^x$
This equation is in the form $y = a \cdot b^x$ where $a = 1$ and $b = 1.5$ , which fits the definition of an exponential equation.
Solution for D: Exponential

Analyze equation E: $g \approx 2(5)^2$
This equation is a constant because $5^2$ is a constant and multiplying by 2 does not change that. It is neither linear nor exponential.
Solution for E: Neither

Analyze equation F: $y = 5(x - 4)^2$
This equation is a quadratic equation, not linear or exponential, because it contains a squared term $(x - 4)^2$ .
Solution for F: Neither

Analyze equation G: $y = 2 - 5(0.6)^x$
This equation is in the form $y = a - b^x$ where $a = 2$ and $b = 0.6$ , which fits the definition of an exponential equation.
Solution for G: Exponential

Analyze equation H: $y = \sqrt{4x + 3}$
This equation involves a square root and is neither linear nor exponential.
Solution for H: Neither

Analyze equation I: $y = 17 - 5x$
This equation is in the form $y = mx + b$ where $m = -5$ and $b = 17$ , fitting the definition of a linear equation.
Solution for I: Linear

Analyze equation J: $y = 7x + 18x^3$
This equation contains a cubic term $18x^3$ and is neither linear nor exponential.
Solution for J: Neither

Analyze equation K: $y = (1.23)^x$
This equation is in the form $y = b^x$ where $b = 1.23$ , fitting the definition of an exponential equation.
Solution for K: Exponential

Analyze equation L: $y = (x + 4)^2 + 3$
This equation is a quadratic equation, not linear or exponential, because it contains a squared term $(x + 4)^2$ .
Solution for L: Neither

Analyze equation M: $y = 7 - 5x$
This equation is in the form $y = mx + b$ where $m = -5$ and $b = 7$ , fitting the definition of a linear equation.
Solution for M: Linear

Analyze equation N: $y = 4x + 5x^2$
This equation contains a quadratic term $5x^2$ and is neither linear nor exponential.
Solution for N: Neither

Analyze equation O: $y = 5(2.4)^x$
This equation is in the form $y = a \cdot b^x$ where $a = 5$ and $b = 2.4$ , fitting the definition of an exponential equation.
Solution for O: Exponential

Analyze equation P: $y = 2x - 4^x$
This equation contains both a linear term $2x$ and an exponential term $-4^x$ , so it does not fit neatly into either category.
Solution for P: Neither