To determine whether each equation is in standard form, we first need to understand what standard form is. In the context of linear equations, standard form is typically written as:
[tex]\[ Ax + By = C \][/tex]
where [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex] are constants, and [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are variables.
Let’s examine each equation step-by-step to see if it fits this format.
1. Equation [tex]\( y = 3x - 5 \)[/tex]
- Rearranging to match the standard form:
[tex]\[ y - 3x = -5 \][/tex] or equivalently [tex]\[-3x + y = -5 \][/tex]
- This fits the [tex]\( Ax + By = C \)[/tex] format.
- Category: Standard Form
2. Equation [tex]\( x = 3 \)[/tex]
- This equation does not involve [tex]\( y \)[/tex], and thus cannot be written in the form [tex]\( Ax + By = C \)[/tex].
- Category: Not Standard
3. Equation [tex]\( 2.7x + 4.1y = -6.8 \)[/tex]
- This equation is already in the form [tex]\( Ax + By = C \)[/tex].
- Category: Standard Form
4. Equation [tex]\( -x - y = 9 \)[/tex]
- This equation is already in the form [tex]\( Ax + By = C \)[/tex] where [tex]\( A = -1 \)[/tex], [tex]\( B = -1 \)[/tex], and [tex]\( C = 9 \)[/tex].
- Category: Standard Form
5. Equation [tex]\( 2x + y = -1 \)[/tex]
- This equation is already in the form [tex]\( Ax + By = C \)[/tex].
- Category: Standard Form
6. Equation [tex]\( 8y - 5 = x \)[/tex]
- Rearranging to match the standard form:
[tex]\[ 8y - x = 5 \][/tex] or equivalently [tex]\[-x + 8y = 5 \][/tex]
- This fits the [tex]\( Ax + By = C \)[/tex] format.
- Category: Standard Form
Let’s summarize our findings:
Standard Form:
- [tex]\( y = 3x - 5 \)[/tex]
- [tex]\( 2.7x + 4.1y = -6.8 \)[/tex]
- [tex]\( -x - y = 9 \)[/tex]
- [tex]\( 2x + y = -1 \)[/tex]
- [tex]\( 8y - 5 = x \)[/tex]
Not Standard:
- [tex]\( x = 3 \)[/tex]
Thus, the sorted equations are:
Standard Form:
[tex]\[ y = 3x - 5 \][/tex]
[tex]\[ 2.7x + 4.1y = -6.8 \][/tex]
[tex]\[ -x - y = 9 \][/tex]
[tex]\[ 2x + y = -1 \][/tex]
[tex]\[ 8y - 5 = x \][/tex]
Not Standard:
[tex]\[ x = 3 \][/tex]
