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The number of years since Keith graduated from middle school can be represented by the equation
y=a-14, where
y is the number of years and
a is his age. Is the relationship between the number of years since Keith graduated and his age proportional or nonproportional?

The number of years since Keith graduated from middle school can be represented by the equation $y=a-14$ , where $y$ is the number of years and $a$ is his age. Is the relationship between the number of years since Keith graduated and his age proportional or nonproportional?

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Find a value using two-variable equations: word problems

Full solution

Q. The number of years since Keith graduated from middle school can be represented by the equation

y=a-14

, where

y

is the number of years and

a

is his age. Is the relationship between the number of years since Keith graduated and his age proportional or nonproportional?

Understand and Identify Variables: Understand the equation and identify the variables. $\newline$ The equation given is $y = a - 14$ , where $y$ represents the number of years since Keith graduated from middle school, and $a$ represents his current age.
Compare to Standard Form: Compare the equation to the standard form of a linear equation. $\newline$ The standard form of a linear equation is $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept. In the given equation, $m$ is $1$ (since there is no coefficient written before $a$ , it is understood to be $1$ ), and $b$ is $-14$ .
Determine Proportional Relationship: Determine if the relationship is proportional. A proportional relationship occurs when the ratio $\frac{y}{x}$ is constant for all values of $x$ and $y$ , and the $y$ -intercept ( $b$ ) is $0$ . In this case, since $b$ is $-14$ and not $0$ , the relationship is not proportional.
Show Ratio is Not Constant: Show that the ratio $y/a$ is not constant. $\newline$ To show that the ratio $y/a$ is not constant, we can substitute different values for $a$ and calculate $y$ each time. $\newline$ Let's take two different ages for Keith, say $a = 20$ and $a = 30$ . $\newline$ For $a = 20$ : $y = 20 - 14 = 6$ $\newline$ For $a = 30$ : $y = 30 - 14 = 16$ $\newline$ Now, calculate the ratios: $\newline$ For $a = 20$ : $y/a$ $1$ $\newline$ For $a = 30$ : $y/a$ $3$ $\newline$ Since the ratios $y/a$ $4$ and $y/a$ $5$ are not equal, the ratio $y/a$ is not constant.

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