Final answer:
Linear equations represent straight lines on a graph and are written in the form y = mx + b, with 'm' being the constant slope and 'b' the y-intercept. Examples include y = 2x + 6 and y = −2x − 3. The y-intercept and slope determine the position and angle of the line, respectively.
Step-by-step explanation:
The subject of the question is linear equations, which are equations of a straight line and can be written in the form y = mx + b, where 'm' is the slope and 'b' is the y-intercept. Considering several examples, let's define linear equations and explore their properties.
For any linear equation, the graph will be a straight line, and the slope (rise over run) will be constant across the entire line. The y-intercept is the point where the line crosses the y-axis. Linear equations can be simple, such as y = 2x + 3, or can represent more complex relationships in real-world scenarios, like the line of best fit y = -173.5 + 4.83x for a data set or predicting the number of flu cases by year. In each case, the structure of the equation follows the linear form and defines a relationship between two variables: the dependent variable 'y' and the independent variable 'x'.
To identify whether an equation is linear, you can look for a constant slope and a simple y = mx + b format. Using the examples given:
- y = 2x + 6 and y = −2x − 3 are both in the form of y = mx + b, indicating they're linear equations with slopes of 2 and -2, respectively, and y-intercepts of 6 and -3.
- y = 9 + 3x is another linear equation with a y-intercept of 9 and a slope of 3, as depicted in Figure A1.
In summary, linear equations represent straight lines on a graph with constant slopes and a y-intercept where they cross the y-axis.