Let's tackle the questions one by one in detail:
### 1. Which of the given equations are linear?
The given equations that are linear are:
- [tex]\( r - 25 = 0 \)[/tex]
- [tex]\( 6p - q = 10 \)[/tex]
- [tex]\( 2s + 3t = -7 \)[/tex]
- [tex]\( c = 12n - 5 \)[/tex]
### 2. How do you describe linear equations?
Linear equations are equations of the first degree, meaning each term is either a constant or the product of a constant and a single variable. In other words, the variables in linear equations are raised only to the power of one. Linear equations do not involve products or powers of variables. They can be written in the general form:
[tex]\[ a_1x_1 + a_2x_2 + \ldots + a_nx_n = b \][/tex]
where [tex]\( a_1, a_2, \ldots, a_n \)[/tex] and [tex]\( b \)[/tex] are constants, and [tex]\( x_1, x_2, \ldots, x_n \)[/tex] are variables.
### 3. Which of the given equations are not linear? Why?
The given equations that are not linear are:
- [tex]\( x^2 - 5x + 3 = 0 \)[/tex]
- [tex]\( r^2 = 144 \)[/tex]
- [tex]\( 9r^2 - 25 = 0 \)[/tex]
- [tex]\( t^2 - 7t + 6 = 0 \)[/tex]
#### Reasons why they are not linear:
- Quadratic Terms: Equations such as [tex]\( x^2 - 5x + 3 = 0 \)[/tex], [tex]\( r^2 = 144 \)[/tex], and [tex]\( t^2 - 7t + 6 = 0 \)[/tex] involve variables raised to the power of 2.
- Non-linear Form: The equation [tex]\( 9r^2 - 25 = 0 \)[/tex] involves [tex]\( r^2 \)[/tex], which is a quadratic term and not linear.
### How are these equations different from those which are linear?
Non-linear equations differ from linear equations in several ways:
- Variable Powers: Non-linear equations include variables raised to powers other than one (e.g., [tex]\( x^2 \)[/tex], [tex]\( r^2 \)[/tex]).
- Complex Relationships: They may also involve expressions where variables are multiplied together.
### Common Characteristics of Non-linear Equations:
- They often involve quadratic terms (e.g., [tex]\( x^2 \)[/tex], [tex]\( r^2 \)[/tex]).
- These equations can curve when graphed, as opposed to linear equations that produce straight lines.
- They are not of the form [tex]\( ax + by + cz + \ldots = d \)[/tex] unless variables have powers other than one or are multiplied together.
To summarize:
- The linear equations in the provided set are [tex]\( r - 25 = 0 \)[/tex], [tex]\( 6p - q = 10 \)[/tex], [tex]\( 2s + 3t = -7 \)[/tex], and [tex]\( c = 12n - 5 \)[/tex].
- Linear equations are equations with variables that have only the first power and no products of variables.
- The non-linear equations are [tex]\( x^2 - 5x + 3 = 0 \)[/tex], [tex]\( r^2 = 144 \)[/tex], [tex]\( 9r^2 - 25 = 0 \)[/tex], and [tex]\( t^2 - 7t + 6 = 0 \)[/tex].
- Non-linear equations include terms where variables are raised to powers other than one or are multiplied together, leading to more complex relationships and shapes when graphed.
