Let's analyze each question based on the given equations:
1. Which of the given equations are linear?
- A linear equation is an equation in which the highest power of the variable is 1. In other words, the variable(s) in the equation appear only to the first power and are not multiplied by each other.
Given equations:
- [tex]\( x^2 - 5x + 3 = 0 \)[/tex]
- [tex]\( 9r^2 - 25 = 0 \)[/tex]
- [tex]\( \frac{1}{2} x^2 + 3x = 8 \)[/tex]
- [tex]\( 8k - 3 = 12 \)[/tex]
- [tex]\( 6p - q = 10 \)[/tex]
- [tex]\( \frac{3}{4} h + 6 = 0 \)[/tex]
From these, the linear equations are:
- [tex]\( 8k - 3 = 12 \)[/tex]
- [tex]\( 6p - q = 10 \)[/tex]
- [tex]\( \frac{3}{4} h + 6 = 0 \)[/tex]
How do you describe a linear equation?
- A linear equation is characterized by the fact that each term is either a constant or the product of a constant and a single variable raised to the first power. Linear equations do not have variables multiplied together (like [tex]\( xy \)[/tex]) or variables raised to a power other than one (like [tex]\( x^2 \)[/tex]).
2. Which of the given equations are not linear? Why?
- The equations that are not linear are:
- [tex]\( x^2 - 5x + 3 = 0 \)[/tex]
- [tex]\( 9r^2 - 25 = 0 \)[/tex]
- [tex]\( \frac{1}{2} x^2 + 3x = 8 \)[/tex]
Why?
- These equations are not linear because they contain variables raised to a power greater than one:
- In [tex]\( x^2 - 5x + 3 = 0 \)[/tex], the term [tex]\( x^2 \)[/tex] indicates it is a quadratic equation.
- In [tex]\( 9r^2 - 25 = 0 \)[/tex], the term [tex]\( r^2 \)[/tex] indicates it is a quadratic equation.
- In [tex]\( \frac{1}{2} x^2 + 3x = 8 \)[/tex], the term [tex]\( x^2 \)[/tex] also indicates it is a quadratic equation.
3. How are these equations different from those which are linear? What common characteristics do these equations have?
- Differences from linear equations:
- The primary difference is that non-linear equations have variables that are raised to a power other than one, such as squared ([tex]\( x^2 \)[/tex] or [tex]\( r^2 \)[/tex]).
- Non-linear equations can also include variables that are multiplied together (though this specific example set does not have such terms).
Common characteristics of non-linear equations:
- They have at least one term in which a variable is raised to a power greater than one.
- They do not graph as straight lines; instead, they can graph as curves such as parabolas, circles, ellipses, or other complex shapes.
- Solutions to non-linear equations typically involve more complex methods than simple algebraic manipulations, often requiring factoring, completing the square, or using the quadratic formula.
In summary, linear equations strictly maintain variables to the first degree and create straight-line graphs, while non-linear equations involve higher degrees or products of variables, resulting in more complex graphical representations.
