Fractions and Decimals

Discrete Mathematics

Differential Equations

Solved on Dec 08, 2023

Determine if the relationship $y=5x(x+3)$ is linear. Explain your reasoning.

STEP 1

Assumptions
1. We are given the relationship $y = 5x(x + 3)$ .
2. We need to determine if this relationship is linear.
3. A linear relationship can be represented by an equation of the form $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.
4. A linear equation graphs to a straight line.
5. Non-linear relationships do not graph to a straight line and often involve powers of $x$ other than 1.

STEP 2

Examine the given relationship.
$y = 5x(x + 3)$

STEP 3

Expand the relationship to see if it simplifies to a linear form.
$y = 5x^2 + 15x$

STEP 4

Analyze the expanded form of the equation.
The equation includes a term with $x^2$ , which is a second-degree term, and a term with $x$ , which is a first-degree term.

STEP 5

Compare the expanded form to the standard form of a linear equation.
A linear equation should only have terms with $x$ to the first power (or constants). The presence of the $x^2$ term indicates that the equation is not linear.

STEP 6

Conclude whether the relationship is linear or not.
Since the equation includes a second-degree term ( $x^2$ ), it is not a linear relationship. It is a quadratic relationship because the highest power of $x$ is 2.
The relationship $y = 5x(x + 3)$ is not linear because it does not fit the form $y = mx + b$ and it includes a quadratic term, $x^2$ .

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