Question

when converting the slope-intercept form $y = mx + b$ to standard form, the terms on one side must be \underline{\hspace{3cm}} to eliminate the $y$ on the right side.

the equation $ax + by = c$ is balanced by ensuring that $a$, $b$, and $c$ are \underline{\hspace{3cm}} without fractions or decimals.

an equation is fully in standard form when $a$ is a positive integer, and $b$ and $c$ are \underline{\hspace{3cm}}.

Explanation:

Step1: Convert slope-intercept to standard form

Start with \(y = mx + b\). To eliminate \(y\) on the right, subtract \(mx + b\) from both sides: \(y - mx - b = 0\), rearranged to \(-mx + y = b\) (preliminary standard form).

Step2: Define balanced standard form

Standard form \(Ax + By = C\) requires \(A, B, C\) to be integers (no fractions/decimals) to be balanced.

Step3: Final standard form conditions

Full standard form needs \(A > 0\) (positive integer), and \(B, C\) as integers (consistent integer requirement).

Answer:

1. Subtract \(mx + b\) from both sides
2. Integers
3. Integers