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Are the linear equations in standard form? Sort each equation into the appropriate category.

Standard Form:
- [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 Form:
- [tex]\(y = 3x - 5\)[/tex]
- [tex]\(x = 3\)[/tex]

Sagot :

GinnyAnswer
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]
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