To determine whether the equation [tex]\( y = x + 3 \)[/tex] is linear or non-linear, we need to analyze its form and properties:
1. General Form of a Linear Equation:
The general form of a linear equation in two variables is given by [tex]\( y = mx + b \)[/tex], where:
- [tex]\( y \)[/tex] is the dependent variable.
- [tex]\( x \)[/tex] is the independent variable.
- [tex]\( m \)[/tex] is the slope of the line.
- [tex]\( b \)[/tex] is the y-intercept.
2. Identification of Components:
- In the equation [tex]\( y = x + 3 \)[/tex]:
- The term [tex]\( x \)[/tex] is the independent variable.
- The coefficient of [tex]\( x \)[/tex] is implicitly [tex]\( 1 \)[/tex], which represents the slope ([tex]\( m = 1 \)[/tex]).
- The constant term [tex]\( 3 \)[/tex] is the y-intercept ([tex]\( b = 3 \)[/tex]).
3. Verification of Linearity:
- For an equation to be linear, it must produce a straight line when graphed.
- It should contain the first power of [tex]\( x \)[/tex] (i.e., [tex]\( x^1 \)[/tex]) without any exponents higher than 1 or any products of variables (like [tex]\( x^2 \)[/tex], [tex]\( xy \)[/tex], etc.).
Given equation [tex]\( y = x + 3 \)[/tex]:
- [tex]\( x \)[/tex] is raised to the power of 1.
- There are no higher powers of [tex]\( x \)[/tex] or products of different variables.
4. Conclusion:
Since the equation [tex]\( y = x + 3 \)[/tex] is in the form [tex]\( y = mx + b \)[/tex] with [tex]\( m = 1 \)[/tex] and [tex]\( b = 3 \)[/tex], it satisfies all the conditions of being a linear equation.
Therefore, the equation [tex]\( y = x + 3 \)[/tex] is linear.
