Linear expressions with more than one variable

Look at the expression $5x-3y+8$ and find the term that contains $y$. That term is $-3y$.

The coefficient is the number multiplied by the variable. In $-3y$, the number multiplied by $y$ is $-3$. The sign belongs to the term!

The coefficient of $y$ is $-3$. It is negative because the term appears as $-3y$ (subtracted), not $+3y$.

We need the coefficient of $y$ — the number multiplied by $y$ in the expression $5x-3y+8$.

$5$ is the coefficient of $x$, not $y$. This option confuses the two different variables.

$8$ is the constant term — it has no variable attached. It is not a coefficient.

$3$ ignores the negative sign. Since the expression reads $-3y$, the sign belongs to the term, making the coefficient $-3$, not $3$.

The only remaining option, $-3$, is correct. The coefficient of $y$ is $-3$.

A linear expression has each variable raised to the power of 1 only. There are no exponents greater than 1, no square roots, and no variables multiplied together.

$3x^2+2y$ has $x^2$ (power of 2 — not linear). $xy+5$ has two variables multiplied together (not linear). $\sqrt{x}+3y$ has a square root of a variable (not linear).

$4x+7y-2$ has $x$ and $y$ each raised to the power of 1, with no square roots or products of variables. This is a linear expression.

We need all variables at power 1, no square roots, and no variables multiplied by each other.

This contains $x^2$, which means $x$ is raised to the power of 2. This violates the definition of a linear expression.

This contains $xy$, which is two variables multiplied together. Linear expressions cannot have variables multiplied by each other.

$\sqrt{x}$ is equivalent to $x^{1/2}$, which means $x$ is raised to a power other than 1. This is not linear.

$4x+7y-2$ is the only expression where every variable appears with power 1. It is the linear expression.